a tetrachotomy for expansions of the real

A TETRACHOTOMY FOR EXPANSIONS OF THE REAL

ORDERED ADDITIVE GROUP

PHILIPP HIERONYMI AND ERIK WALSBERG

Abstract. Let R be an expansion of the ordered real additive group. When

R is o-minimal, it is known that either R defines an ordered field isomorphicto (R, <,+, ·) on some open subinterval I ⊆ R, or R is a reduct of an ordered

vector space. We say R is field-type if it satisfies the former condition. In this

paper, we prove a more general result for arbitrary expansions of (R, <,+).In particular, we show that for expansions that do not define dense ω-orders

(we call these type A expansions), an appropriate version of Zilber’s principle

holds. Among other things we conclude that in a type A expansion that is notfield-type, every continuous definable function [0, 1]m → Rn is locally affine

outside a nowhere dense set.

1. Introduction

A classical theme in model theory, dating back to Zilber’s trichotomy conjec-ture [41], is to analyze whether model-theoretically tame structures that exhibitwell-defined non-linear behavior, actually define fields. While Zilber’s original con-jecture has famously been proven false by Hrushovski [27], Peterzil and Starchenko[37] were able to show that for o-minimal structures non-linearity yields a defin-able field. Restricting ourselves to expansions of the real ordered additive group,we produce in this paper a vast generalization of this result and earlier results ofMarker, Peterzil and Pillay [34].

Throughout this paper, fix a first-order expansion R = (R, <,+, . . .) of the or-dered additive group of real numbers. Definable without modification means“R-definable, possibly with parameters”, and I, J, L always range over nonemptybounded open subintervals of R. We say R is field-type if there are definablefunctions ⊕,⊗ : I2 → I such that (I,<,⊕,⊗) is an ordered field. Since (I,<) isDedekind complete, we have that (I,<,⊕,⊗) is isomorphic to (R, <,+, ·). It iseasy to see that ⊕,⊗ must be continuous. We let RVec be the ordered R-vectorspace (R, <,+, (x 7→ λx)λ∈R).

It follows from [34] and Loveys and Peterzil [33] that when R is o-minimal, the ex-pansion R is either field-type or is a reduct of the ordered vector space RVec. Thus

Date: October 17, 2021.2010 Mathematics Subject Classification. Primary 03C64 Secondary 03C40, 03D05, 03E15,

26A21, 54F45.The first author was partially supported by NSF grants DMS-1300402 and DMS-1654725. The

second author was partially supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/

MODAG. A preprint of this paper was disseminated under the title “On continuous functions

definable in expansions of the ordered real additive group”.

1

2 PHILIPP HIERONYMI AND ERIK WALSBERG

for o-minimal expansions of (R, <,+) a strong dichotomy into linear and field-typestructures is already known. In order to prove corresponding results for arbitraryexpansions of (R, <,+), we need to split the collection of all such expansions intotwo parts, and then prove Zilber-style dichotomy results separating linear and non-linear behavior for both parts. This results in a tetrachotomy of all expansionsvisualized in Figure 1. The criterion we use to separate all expansions of (R, <,+)is whether or not they define a dense ω-order. An ω-orderable set (or short: anω-order) is a definable set that is either finite or admits a definable ordering withorder type ω. We say such a set is dense if it is dense in some nonempty opensubinterval of R. The behavior of definable sets and functions in R largely dependson whether or not R admits a dense ω-order. We say that R is type A if it doesnot admit a dense ω-order. We say R is type C if it defines every compact set andtype B if it admits a dense ω-order and is not type C.

Before discussing questions of linearity versus non-linearity, we want to give a briefrationale for dividing expansions into the types A, B and C. First note that a typeC expansion is as wild as can be from a model-theoretic standpoint. Indeed, everyprojective subset of [0, 1]n in the sense of the projective hierarchy from descriptiveset theory (see Kechris [29, 37.6]), in particular every Borel function on a boundeddomain, is definable in a type C structure. Even the question whether all definablesets in a fixed type C expansion are Lebesgue measurable is independent of ZFC.From a combinatorical model-theoretic point of view, type B expansions are notmuch better: by [26, Theorem B] every type B expansion defines an isomorphiccopy of the standard model (P(N),N,∈,+1) of the monadic second order theory of(N,+1), where P(N) is the power set of N. It follows that a type B expansion cannotsatisfy any Shelah-style model-theoretic tameness property such as NIP or NTP2

(see e.g. Simon [40] for definitions). It is also easy to see that type B expansionsdo not satisfy any of the classical logical-geometric tameness notions such as o-minimality, local o-minimality, or d-minimality. Thus all the usual model-theoretic

dense ω-order

field-type

no yes

yes

no

type C

Every continuous functionis definable.

type A, field-type

For any k, every definable

continuous function is Ck

almost everywhere.

type A, affine

Every definable continuousfunction is affine almost

everywhere.

type B

Every definable C2 functionis affine.

Figure 1. A tetrachotomy: Defining a dense ω-order separatestame and wild expansions, being of field-type distinguishes betweenlinear and non-linear expansions.

A TETRACHOTOMY 3

and geometric tameness notions in the literature imply type A. The noteworthyexception is decidability of the theory. Examples of type B expansions with de-cidable theories are (R, <,+, x 7→ αx,Z) where α ∈ R is a quadratic irrational(see [23]) and (R, <,+, C), where C is the middle-thirds Cantor set (see Balder-rama and Hieronymi [4]). We describe another interesting example in Section 8.3.Type B expansions have received little attention within tame geometry and modeltheory, but have appeared in theoretical computer science (Boigelot, Rassart, andWolper [9]) and fractal geometry (Charlier, Leroy, and Rigo [11]). One reason mightbe that all known examples of type B expansions are mutually interpretable with(P(N),N,∈,+1). The theory of the latter structure was shown to be decidable byBuchi [10] using automata-theoretic rather then model-theoretic methods.

Returning to the question of linearity of a given expansion, observe that every typeC expansion is field-type. On the other hand, type B expansions are known tonot be field-type, and we will derive various results in this paper that clarify thelinearity of such structures. Arguably the main contribution of this paper is thefollowing Zilber-style dichotomy theorem for type A expansions.

1.1. A dichotomy for type A expansions. Before we state the result, we haveto introduce some notation. A set X ⊆ Rn is DΣ if there is a definable family {Yr,s :r, s > 0} of compact subsets of Rn such that X =

⋃r,s Yr,s, and Yr,s ⊆ Yr′,s′ if r ≤ r′

and s ≥ s′, for all r, r′, s, s′ > 0. A function is DΣ if its graph is DΣ, and a definablefield (X,⊕,⊗) is DΣ whenever X and ⊕,⊗ are DΣ. All open and closed definablesets are DΣ and the collection of DΣ-sets is closed under finite intersections, finiteunions, cartesian products, and images under continuous definable functions. Agood theory of dimension, definable selection, and generic smoothness are enoughto obtain many results in the o-minimal setting. By Fornasiero et al. [18] the firsttwo also hold in the type A setting for DΣ sets, and here we obtain the third. Thisallows us to extend the well-known results in [37] from the o-minimal to the moregeneral type A setting.

Theorem A. Suppose R is type A. Then the following are equivalent:

(1) R is field-type,(2) there is a DΣ field (X,⊕,⊗) with dimX > 0,(3) there is a DΣ family (Ax)x∈B of subsets of Rn such that dimB ≥ 2, each Ax

is one-dimensional, and Ax ∩Ay is zero-dimensional for distinct x, y ∈ B,(4) there is a definable open U ⊆ Rm and a DΣ function f : U → Rn that is

nowhere locally affine.

In particular, if R is not field-type, then every continuous definable function U →Rn, where U is a definable open subset of Rm, is locally affine outside a nowheredense subset of U .

The equivalence of (1) and (3) essentially states that a version of Zilber’s principle(as stated in [37, Definition 1.6]) holds for type A expansions. If R is o-minimal,then every definable set is DΣ. Thus Theorem A indeed generalizes the result fromthe o-minimal setting.

There are type A expansions that are not field-type, yet define infinite fields. Forexample, the expansion of (R, <,+) by all subsets of all Zn is locally o-minimal andhence type A (this follows from either Kawakami et al.[28] or Friedman and Miller


[20]). Such pathological examples give an upper limit on what can be proven in thegeneral setting of type A expansions.

As pointed out above, an essential tool we need for the proof of Theorem A isgeneric Ck-smoothness for DΣ-functions in type A expansions. Letting U be adefinable open subset of Rm, we say that a property holds almost everywhere,or generically, on U if there is a dense definable open subset of U on which itholds. It follows from [18, Theorem D] that if R is type A, then a nowhere densedefinable subset of Rk has null k-dimensional Lebesgue measure. So if a propertyholds almost everywhere in our sense, it holds almost everywhere in the usualmeasure-theoretic sense.

Theorem B. Suppose that R is type A. Fix k ≥ 0. Let U be a definable open subsetof Rm and f : U → Rn be DΣ. Then f is generically Ck on U . In particular, everycontinuous definable f : U → Rn is generically Ck.

Thus, if U ⊆ Rm is open and f : U → Rn is continuous and not generically Ck

for some k ≥ 1, then (R, <,+, f) defines an isomorphic copy of (P(N),N,∈,+1).Laskowski and Steinhorn [31] prove Theorem B in the o-minimal setting. Ourproof makes crucial use of their ideas. In particular, we also use a classical theo-rem of Boas and Widder [7]. Theorem B fails for arbitrary definable functions, as(R, <,+,Q) is type A and the characteristic function of Q is nowhere continuous.Further observe that Theorem B cannot be strengthened to assert that a continu-ous function definable in a type A expansion is generically C∞. Such a result failsalready in the o-minimal setting by Rolin, Speissegger, and Wilkie [39]. In the caseof expansions of (R, <,+, ·), the one variable case of Theorem B is due to Fornase-rio [17]. However, this special case is substantially easier because of definability ofdivision.

In order to prove Theorem A, we need to combine Theorem B with the followingresult essentially due to Marker, Peterzil, and Pillay [34, Section 3].

Fact 1.1. If R defines a C2 non-affine function I → R, then R is field-type.

Their proof is only written to cover the case when R is o-minimal, but goes throughin general (see our proof of Theorem F below). We first prove the one-variable caseof Theorem B, apply this to prove Theorem A, and then apply Theorem A to provethe multivariable case of Theorem B.

We believe that the study of type A expansions is the ultimate generalization ofo-minimality in the setting of expansions of (R, <,+). Introduced by Miller andSpeissegger [35], the open core R◦ of R is the expansion of (R, <) by all openR-definable subsets of all Rn. We hope to eventually show that if R is type A,then R◦-definable sets and functions behave in a similar fashion to those definablein o-minimal expansions. At present we are confined to the collection of DΣ sets.

1.2. Linearity of type B expansions. We now discuss the case when R admitsa dense ω-order. The key result from [22] can be restated as follows1.

1To prove Fact 1.2, the reader can either easily redo the proof of [22, Theorem 1.1] or simplyapply [18, Theorem C].

A TETRACHOTOMY 5

Fact 1.2. Suppose R expands (R, <,+, ·). Then there is a dense ω-order if andonly if R defines the set of integers.

We immediately obtain the following corollary of Fact 1.2.

Fact 1.3. Suppose R admits a dense ω-order. Then R is field-type if and only ifR is type C.

Thus studying the linearity of expansions that admit a dense ω-order and are notfield-type, reduces to studying the linearity of type B expansions.

To capture this linearity, we introduce the notion of a weak pole. Recall that apole is a continuous surjection from a bounded interval to an unbounded interval.A weak pole is a definable family {hd : d ∈ E} of continuous maps hd : [0, d]→ Rsuch that

(i) E ⊆ R>0 is closed in R>0 and (0, ε) ∩ E 6= ∅ for all ε > 0,(ii) there is δ > 0 such that [0, δ] ⊆ hd([0, d]) for all d ∈ E.

It is easy to see that if R is field-type, then R admits a weak pole. Our first resultis the following strengthening of Fact 1.3.

Theorem C. Suppose R admits a dense ω-order. Then R is type C if and only ifit defines weak pole.

Thus a type B expansion cannot define a weak pole. Structures that do not defineweak poles, independently of whether they define a dense ω-order, exhibit linearbehavior.

Theorem D. Suppose R does not admit a weak pole. Then

(1) If W ⊆ Rm is open and bounded, then every continuous definable functionf : W → Rn is bounded.

(2) Every continuous definable function R>0 → R is bounded above by an affinefunction.

(3) Every definable family of linear functions [0, 1]→ R has only finitely manydistinct elements.

Note that Theorem D shows that if R admits a pole, then R admits a weak pole.The converse does not always hold. The expansion of (R, <,+) by all boundedsemialgebraic sets clearly admits a weak pole, but does not admit a pole by Pillay,Scowcroft and Steinhorn [38]. It follows from the quantifier elimination for RVec

that RVec does not admit a weak pole. Thus if R is o-minimal, then R admitsa weak pole if and only if R is field-type. However, there are type A expansionsthat admit weak poles and are not field-type. Let g : 2Z × R → R be given byf(t, t′) = tt′. Then (R, <,+, g) is a reduct of (R, <,+, ·, 2Z) and is therefore typeA by van den Dries [14]. Delon [12] studied (R, <,+, g). It can be deduced from[12, Theorem 2] that (R, <,+, g) is not field-type. For t ∈ 2Z, let gt : [0, 1]→ R begiven by gt(t

′) = tt′. It is easy to see that {gt : t ∈ 2Z} is a weak pole.

We also show that continuous definable functions I → R in type B expansions satisfyanother important property of affine functions. We say f : I → R is repetitious iffor every open subinterval J ⊆ I there are δ > 0, x, y ∈ J such that δ < y − x and

f(x+ ε)− f(x) = f(y + ε)− f(y) for all 0 ≤ ε < δ


dense ω-order

field-type

no yes

yes

no

type C

type A, field-type

type A, affine

type B

o-m

inim

al

Figure 2. The stripped area indicates expansions that do notdefine weak poles. An o-minimal structure defines a weak pole ifand only if it is field-type.

Affine functions are obviously repetitious. However, a strictly convex function I →R is not repetitious.

Theorem E. Suppose R is type B. Then every continuous definable function I → Ris repetitious.

Thus, if R defines a continuous function I → R that is not repetitious, then R isfield-type. A special case is that if R defines a strictly convex function I → R,then R is field-type. While most familiar examples of continuous nowhere locallyaffine functions are not repetitious, there are continuous repetitious functions thatare nowhere locally affine. If f = (f1, f2) : [0, 1] → [0, 1]2 is the classical Hilbertspace-filling curve, then one can check that f1 and f2 are repetitious.

It is natural to ask if a type B expansion can interpret an infinite field. By AbuZaid, Gradel, Kaiser, and Pakusa [2] the structure (P(N),N,∈,+1) does not ad-mit a parameter-free interpretation of an infinite field. Abu Zaid [1] shows that(P(N),N,∈,+1) does not interpret (R, <,+, ·), but it appears to be an open ques-tion whether (P(N),N,∈,+1) interprets an infinite field.

1.3. Open questions. Despite the results above we do not know the full extentto which type B expansions are linear. In particular, we do not know the answerto the following question.

Question 1.4. Suppose that R is type B. Let f : I → R be continuous and definable.Is f generically locally affine?

Block Gorman et al. [6] give a positive answer to this question for certain naturaltype B expansions, but the automata-theoretic argument in that paper is unlikely toextend to all type B expansions. By Theorem A this question has a positive answerfor type A expansions that are not field-type. Thus a positive answer to Question1.4 would show that all expansions that are not field-type, satisfy this strong formof linearity. Question 1.4 can be stated in three non-trivially equivalent forms.

A TETRACHOTOMY 7

Corollary 1.5. Let f : I → R be continuous and definable. The following state-ments are equivalent:

(1) If R is type B, then f is generically locally affine.(2) If f is nowhere locally affine, then R is field-type.(3) If f is nowhere Ck for some k ≥ 2, then R is type C.

Proof. Theorem B and Fact 1.1 show that (1) implies (2). Since every nowhereCk function is also nowhere locally affine, the implication (2) ⇒ (3) follows fromTheorem B and Fact 1.3. Since every C2-function definable in a type B expansionis affine, we see that (3) implies (1). �

Note that statement (2) neither refers to type B or C nor to ω-orders, but ratherasks from what kind of objects we can recover a field. We regard this as one of themain open questions in the study of expansions of (R, <,+).

Another natural question is whether Fact 1.1 can be extended to C1 functions.While we are unable to answer this question in general, we give a positive answerunder an extremely weak model-theoretic assumption.

Theorem F. Suppose R does not define an isomorphic copy of the standard model(P(N),N,∈,+, ·) of second order arthimetic. If R defines a non-affine C1 functionf : I → R, then R is field-type.

Thus every C1 function I → R definable in a type B structure with a decidabletheory is affine. This covers the examples of type B structures described above.

1.4. Applications. We anticipate that applications of the results presented in thispaper are numerous. In Section 8 we already collect the most immediate conse-quences of our work related to descriptive set theory and automata theory. Whilethese results are interesting in their own right, we do not wish to further extendthe introduction. We refer the reader to Section 8 for a precise description of theseresults.

Acknowledgements. We thank Samantha Xu for useful conversations on thetopic of this paper, Kobi Peterzil for answering our questions related to [34], andChris Miller and Michel Rigo for correspondence around Section 8. We thank theanonymous referee for very helpful comments that improved the presentation ofthis paper.

Notations. Let X ⊆ Rn. We denote by Cl(X) the closure of X, by Int(A) the inte-rior of X, and by Bd(X) the boundary Cl(X)\ Int(X) of X. Whenever X ⊆ Rm+n

and x ∈ Rm, then Xx denotes the set {y ∈ Rn : (x, y) ∈ X}. A box is a subsetof Rk given as a product of k nonempty open intervals.

We always use i, j, k, l,m, n,N for natural numbers and r, s, t, λ, ε, δ for real num-bers. We let ‖x‖ := max{|x1|, . . . , |xn|} be the l∞ norm of x = (x1, . . . , xn) ∈ Rn.

2. Preliminaries

2.1. DΣ sets in type A expansions. A set X ⊆ Rn is DΣ if there is a definablefamily {Yr,s : r, s > 0} of compact subsets of Rn such that X =

⋃r,s Yr,s, and

Yr,s ⊆ Yr′,s′ if r ≤ r′ and s ≥ s′, for all r, r′, s, s′ > 0. The family {Yr,s : r, s > 0}


witnessses that X is DΣ.

Every DΣ set is Fσ, and this might lead us to think of DΣ sets as “definably Fσ”.Note however that there can be definable Fσ sets that are not DΣ. For exampleFact 2.2 below shows that Q is not a DΣ set in (R, <,+,Q). A function is DΣ ifits graph is DΣ. We say that a family {Ax : x ∈ Rm} of subsets of Rn is DΣ if theset of (x, y) ∈ Rm × Rn satisfying y ∈ Ax is DΣ.

Fact 2.1 (Dolich, Miller and Steinhorn [13]). Open and closed definable subsetsof Rm are DΣ, finite unions and finite intersections of DΣ sets are DΣ, and theimage or pre-image of a DΣ set under a continuous definable function is DΣ. Inparticular, a continuous definable function whose domain is either an open or closedsubset of Rm is DΣ.

If R is o-minimal, then every definable set is a boolean combination of closeddefinable sets by cell decomposition. So in this situation every definable set is DΣ.In general, the complement of a DΣ set is not DΣ. In this paper, we will makeextensive use of the Strong Baire Category Theorem, or SBCT, establishedin [18].

Fact 2.2 (SBCT [18, Theorem D]). Suppose R is type A. Then every DΣ subsetof Rn either has interior or is nowhere dense.

Another result we use repeatedly is the following DΣ-selection result.

Fact 2.3 ([18, Proposition 5.5]). Suppose R is type A. Let A ⊆ Rm+n be DΣ suchthat π(A) has interior, where π : Rm+n → Rm is the projection onto the first mcoordinates. Then there is a definable open subset V of Rm such that V ⊆ π(A)and a continuous definable f : V → Rn such that (p, f(p)) ∈ A for all p ∈ V .

Theorem 2.4 is an easy consequence of Fact 2.3. We leave the details to the reader.

Theorem 2.4. Suppose R is type A. Let U ⊆ Rm be definable open and let f :U → Rn be DΣ. Then f is generically continuous.

For the continuous functions in type A structures the following weak monotonicitytheorem for type A structures.

Fact 2.5 ([26, Theorem 4.3] and [18, Fact 3.3]). Suppose R is type A. Let Z ⊆ Rnbe definable and let (fz : R→ R)z∈Z be a definable family of continuous functions.Then there is a definable family (Uz)z∈Z of open dense subsets of R such that forevery z ∈ Z the function fz is strictly increasing, strictly decreasing, or constanton each connected component of Uz.

The uniformity in the statement of the weak monotonicity theorems is not explicitin the literature, but an inspection of the proof of [26, Theorem 4.3] shows that thedefinable open set U is clearly constructed uniformly in the parameters defining f .

In a few place throughout this paper we will refer to the dimension of a DΣ set in atype A expansion. It is necessary to explain what dimension we refer to. Given X ⊆Rn we let dim(X) be the topological dimension ofX. Topological dimension hererefers to either small inductive dimension, large inductive dimension, or Lebesguecovering dimension. These three dimensions coincide on all subsets of Rn (seeEngelking [16] for details and definitions). Model-theorists usually consider as a

A TETRACHOTOMY 9

dimension of a subset X of Rn the maximal k for which there is a coordinateprojection ρ : Rn → Rk such that ρ(X) has nonempty interior. In [18] this is calledthe naive dimension of X. In general, this naive dimension is not well-behavedfor arbitrary subsets of Rn and does not equal the topological dimension. However,for DΣ sets these notions of dimension coincide.

Fact 2.6 ([18, Proposition 5.7, Theorem F]). Suppose R is type A. Let X ⊆ Rnbe DΣ. Then dim(X) is equal to the maximal k for which there is a coordi-nate projection ρ : Rn → Rk such that ρ(X) has nonempty interior. Moreover,dim Cl(X) = dim(X).

Corollary 2.7 follows from DΣ-selection and Fact 2.6.

Corollary 2.7. Suppose R is type A. Suppose X ⊆ Rn is DΣ and dimX ≥ d. Thenthere is a nonempty definable open U ⊆ Rd and a continuous definable injectionf : U → X.

We also make use of the following.

Fact 2.8 ([18, Theorem F]). Suppose R is type A. Let X ⊆ Rn be DΣ and supposef : X → Rm is definable and continuous. Then dim f(X) ≤ dimX and dim f(X) =dimX when f is a bijection.

Finally, we also need the following consequence of the Kuratowski-Ulam theorem(see [29, Theorem 8.41]).

Fact 2.9. Suppose U ⊆ Rm, V ⊆ Rn are nonempty and open. Let A be an Fσsubset of U × V . If Ax contains a dense open subset of V for all x ∈ U , then Acontains a dense open subset of U × V .

2.2. Expansions that define dense ω-orders. In this section we recall a fewfundamental results about expansions that are not type A. The following is a minormodification of Hieronymi and Tychonievich [25, Theorem A].

Fact 2.10 ([18, Proposition 3.8]). If R defines a linear order (D,≺), an openinterval I ⊆ R, and a function g : R3 ×D → D such that

(i) (D,≺) has order type ω and D is dense in I,(ii) for every a, b ∈ U and e, d ∈ D with a < b and e � d,

{c ∈ R : g(c, a, b, d) = e} ∩ (a, b) has nonempty interior,

then R is type C.

By [26, Theorem B] every expansion of (R, <,+) that defines a dense ω-orderableset, interprets the monadic second order theory of one successor. Fact 2.10 roughlystates that we can not expand R too much without it becoming type C, once wehave this ω-orderable set. This should be compared to a similar result of Elgot andRabin [15, Theorem 1] on decidable expansions of the monadic second theory ofone successor (see [4, Section 3.2] for a more detailed discussion). The followingcorollary to Fact 2.10 is often easier to apply.

Fact 2.11 ([4, Lemma 3.7]). If there exist two dense ω-orderable subsets C and Dof [0, 1] satisfying (C − C) ∩ (D −D) = {0}, then R is type C.


3. Defining a field

In this section we explain how to recover a field from a non-affine function. Inthe case of a C2 function our central argument has already appeared in [34]. Ourmain contribution is the extension to C1 functions, in particular statement (2) inthe following theorem.

Theorem 3.1. Let f : I → R be definable, C1, and non-affine.

(1) If f ′ is strictly increasing or strictly decreasing on some open subintervalof I, then R is field-type.

(2) If f ′ is not strictly increasing or strictly decreasing on any open subintervalof I, then R defines an isomorphic copy of (P(N),N,∈,+, ·).

In particular, if f is C2, then R is field-type.

Theorem F follows. The conclusion of statement (2) in Theorem 3.1 does not ruleout the possibility that R is type A. Note that a structure defines an isomorphiccopy of (P(N),N,∈,+, ·) if and only if it defines an isomorphic copy of (R, <,+, ·,Z).Friedman, Kurdyka, Miller, and Speissegger [19] gave an example of type A struc-ture that defines an isomorphic copy of (R, <,+, ·,Z).

Before proving Theorem 3.1 we establish a few lemmas used in the proof. We fix onefurther notation: A complementary interval of A ⊆ R is a connected componentof the complement of the closure of A.

Lemma 3.2. Let F ⊆ R be such that (F,<) is isomorphic to (R, <). Then F eitherhas interior or is nowhere dense. Furthermore, if I is a bounded complementaryinterval of F , then either the left endpoint or the right endpoint of I is in F .

Proof. Let ι : (R, <) → (F,<) be an isomorphism. Let J be an open interval.We suppose that F is dense in J and show J ⊆ F . The first claim then follows.Let t ∈ J and X := {x ∈ R : ι(x) < t}. The density of F in J yields an x ∈ Rsatisfying ι(x) > t. Thus X is bounded from above. Let u be the supremumof X in R. As F is dense in J , we must have ι(u) = t. Therefore t ∈ F . Weproceed to the second claim. Let I be a bounded complementary interval of F .The density of (F,<) shows that F contains at most one endpoint of I. Let z ∈ Iand Y := {x ∈ R : ι(x) < z}. As I is a bounded complementary interval, there isan x ∈ R such that ι(x) > z. Hence Y is bounded above in R. Let u ∈ R be thesupremum of Y . If u ∈ Y , then ι(u) is the left endpoint of I. If u /∈ Y , then ι(u) isthe right endpoint of I. �

Lemma 3.3. Let A ⊆ R be definable and bounded. Then the set D of endpoints ofbounded complementary intervals of A is ω-orderable.

Proof. The statement trivially holds when D is finite. We now consider the casethat D is infinite, and define an ω-order ≺ on D. Note that each element of D isthe endpoint of at most two bounded complementary intervals. Let δ : D → R bethe definable function that maps each d to the minimal length of a complementaryinterval with endpoint d. We declare d ≺ d′ if either δ(d′) < δ(d) or (δ(d′) = δ(d)and d < d′). It is easy to see that ≺ is an ω-order on D (see Section 2 of [26] fordetails). �

A TETRACHOTOMY 11

Lemma 3.4. Let F ⊆ R be definable and bounded, and ⊕,⊗ : F 2 → F be definablesuch that (F,<,⊕,⊗) is isomorphic to (R, <,+, ·). Then F either has interior oris nowhere dense and,

(1) if F has interior, then R is field-type.(2) if F is nowhere dense, then there is a definable Z ⊆ F such that the struc-

ture (F,<,⊕,⊗, Z) is isomorphic to (R, <,+, ·,Z).

Proof. Lemma 3.2 shows that F either has interior or is nowhere dense. Item (1)above follows easily from the fact that for every open interval I there are (R, <,+, ·)-definable ⊕′,⊗′ : I2 → I such that (I,<,⊕′,⊗′) is isomorphic to (R, <,+, ·). Weleave the details of (1) to the reader and prove (2). Suppose F is nowhere dense. LetD be the set of endpoints of bounded complementary intervals of F and D′ = D∩F .

We first show that D′ is dense in F . Let x, y ∈ F and x < y. Since F is nowheredense, there is a complementary interval I of F such that x < z < y for every z ∈ I.By Lemma 3.2 one of the endpoints of I lies in F . Thus D′ is dense in F .

Let ≺ be the ω-order on D given by Lemma 3.3 and denote its restriction to D′

by ≺′. Note that (D′,≺′) has order-type ω. Consider F := (F,<,⊕,⊗, D′,≺′).Clearly F is definable. Note that ι is an isomorphism between F and an ex-pansion of (R, <,+, ·) that admits a dense ω-orderable set. An application ofFact 1.2 shows that F defines Z := ι−1(Z) and that (F,<,⊕,⊗, Z) is isomorphic to(R, <,+, ·,Z). �

Lemma 3.5. Let f : [a, b]→ R be C1 and definable, let {(gx : [0, cx]→ R) : x ∈ X}be a definable family of functions such that

(1) f ′(a) = 0 and f ′(t) > 0 for all a < t ≤ b, and(2) cx > 0, gx(0) = 0, and g′x(0) exists for all x ∈ X.

Then the relations g′y(0) < g′x(0), g′x(0) ≤ g′y(0), and g′x(0) = g′y(0) are definable onX.

Proof. We only prove the first claim, the latter two follow. Fix x, y ∈ X. We showthat the following are equivalent:

(i) g′x(0) < g′y(0),(ii) there is z ∈ (a, b) such that

(?) gx(ε) + [f(z + ε)− f(z)] < gy(ε) for sufficiently small ε > 0.

Suppose (ii) holds. Let z ∈ (a, b) be such that (?) holds. Dividing by ε and takingthe limit as ε→ 0, we get

g′x(0) + f ′(z) ≤ g′y(0).

As z > a, we have f ′(z) > 0. Thus g′x(0) < g′y(0) and (i) holds.

Suppose (i) holds. Let δ > 0 be such that g′x(0) + δ < g′y(0). Since f ′ is continuousand f ′(a) = 0 < f ′(b), there is z ∈ (a, b) such that f ′(z) < δ. Fix such z. Theng′x(0) + f ′(z) < g′y(0). Thus

limε→0

gx(ε)

ε+ limε→0

f(z + ε)− f(z)

ε< limε→0

gy(ε)

ε.


Hence

gx(ε)

ε+f(z + ε)− f(z)

ε<gy(ε)

εfor sufficiently small ε > 0.

Therefore (?) holds. �

Proof of Theorem 3.1. There are a, b ∈ R with a < b such that f ′(a) 6= f ′(b) andone of the following two cases holds:

(I) f ′ is strictly increasing or strictly decreasing on [a, b],(II) there is no open subinterval of [a, b] on which f ′ is strictly increasing or

strictly decreasing.

Now replace f by its restriction to [a, b]. So from now on, f is a function from [a, b]to R satisfying f ′(a) 6= f ′(b) and either condition (I) or (II). After replacing f with−f if necessary, we suppose that f ′(a) < f ′(b). In case (I) these assumptions implyf ′ is strictly increasing.

Let q ∈ Q be such that f ′(a) < q < f ′(b). By continuity of f ′ there is an x ∈ (a, b)such that f ′(x) = q. Continuity of f ′ further implies that the set of such x is closed.Let c be the maximal element of [a, b] such that f ′(c) = q. Note that c < b. By theintermediate value theorem, f ′(x) > q for all x ∈ (c, b]. After replacing a with c ifnecessary, we may assume that f ′(a) = q and f ′(x) > q for all a < x ≤ b.

Let h : [a, b]→ R be given by h(x) = f(x)− q(x− a). Note that h′(x) = f ′(x)− qfor all x ∈ [a, b]. Thus h′(a) = 0 and h′(x) > 0 for all a < x ≤ b. Moreover, h′ isstrictly increasing if f ′ is. Thus h′ is strictly increasing in case (I). Since q is rational,h is definable. After replacing f with h if necessary, we may suppose that f ′(a) = 0.

Let N ∈ N satisfy f ′(b) ≥ 1N . After replacing f with Nf if necessary, we can

assume f ′(b) ≥ 1. Let d be the minimal element of [a, b] such that f ′(d) = 1. Afterreplacing b with d if necessary, we may suppose that f ′(b) = 1 and 0 < f ′(x) < 1for all a < x < b.

Applying Lemma 3.5 to the definable family gx(t) = f(x+ t)−f(x) we see that therelations f ′(x) < f ′(y), f ′(x) ≤ f ′(y), and f ′(x) = f ′(y) are definable on I. LetE ⊆ [a, b] be the set of x such that f ′(y) < f ′(x) for all y ∈ [a, x). Observe thatE is definable. For every t ∈ [0, 1] the set {z ∈ [a, b] : f ′(z) ≥ t} is closed andnonempty. Therefore this set has a minimal element w. Since f ′(a) = 0 and f ′ iscontinuous, this minimal element w must satisfy f ′(w) = t. In particular, w ∈ E.Thus for every t ∈ [0, 1] there is an x ∈ E such that f ′(x) = t. Note that a, b ∈ E.Furthermore, if x, y ∈ E and x < y, then f ′(x) < f ′(y). It follows that x 7→ f ′(x)gives an isomorphism between (E,<) and ([0, 1], <).

In case (I) we trivially have E = [a, b], because in this case f ′ is strictly increasing.If E contains an open interval, then f ′ must be strictly increasing on that interval.Thus E has empty interior in case (II).

For x ∈ E \{b} and t ∈ [0, b−x] we set fx(t) = f(x+t)−f(x). We declare fb(t) = tfor all t > 0. Then f ′x(0) = f ′(x) for all x ∈ E. As f is strictly increasing, each fxis strictly increasing. We suppose a = 0 after translating [a, b] if necessary. Then

A TETRACHOTOMY 13

E is a subset of [0, b]. We declare

E1 = −(E \ {0, b}) + 2b, E2 = −(E \ {0}), E3 = −E1.

Then E,E1, E2, E3 are pairwise disjoint as they are subsets of [0, b], (b, 2b), [−b, 0),and (−2b,−b) respectively. Set F = E∪E1∪E2∪E3. Note that in case (I) we haveF = (−2b, 2b). So F is an interval in this situation. Furthermore, in case (II) theset F has empty interior as E and each Ei have empty interior. We now constructa definable family of functions {hx : x ∈ F} with the following two properties:

(i) For all t ∈ R there is a unique x ∈ F such that h′x(0) = t.(ii) If x, y ∈ F and x < y, then h′x(0) < h′y(0).

When x ∈ E, we set hx = fx. When x ∈ E1, set hx to be the compositional inverseof f2b−x. Since each fx is strictly increasing, each fx has a compositional inverse.Observe that h′x(0) = f ′2b−x(0)−1 for all x ∈ E1. It follows that for every t > 1there is a unique x ∈ E1 such that h′x(0) = t. When x ∈ E2, we let hx = −f−x.Then h′x(0) = −f ′−x(0) for all x ∈ E2. Again we directly deduce that for everyt ∈ [−1, 0] there is a unique x ∈ E2 such that h′x(0) = t. Finally, if x ∈ E3, weset hx = −h−x. Also in this situation we get that for all t < −1 there is a uniquex ∈ E3 such that h′x(0) = t. Conditions (i) and (ii) above follow.

We are ready to define the field structure on F . For this, we need to define twofunctions ⊕,⊗ : F 2 → F . Given x, y ∈ F , we let x⊕ y be the unique element of Fsuch that

h′x⊕y(0) = (hx + hy)′(0)

and x⊗ y be the unique element of F such that

h′x⊗y(0) = (hx ◦ hy)′(0).

It follows easily from Lemma 3.5 that ⊕ and ⊗ are definable. By our construction,we immediately get that for all x, y ∈ F

h′x⊕y(0) = h′x(0) + h′y(0) and h′x⊗y(0) = h′x(0)h′y(0).

So x 7→ h′x(0) gives an isomorphism (F,<,⊕,⊗)→ (R, <,+, ·). As observed above,F is an interval in case (I) and has empty interior in case (II). Now apply Lemma 3.4.

�

We record some corollaries.

Corollary 3.6. Let f : I → R be a non-affine C1 and generically locally affinefunction. Then

(1) (R, <,+, f) defines an isomorphic copy of (P(N),N,∈,+, ·).(2) (R, <,+, ·, f) is type C.

Proof. The derivative of f is locally constant almost everywhere and therefore isnot strictly increasing or strictly decreasing on any open subinterval of I. Thus(R, <,+, f) defines an isomorphic copy of (P(N),N,∈,+, ·) by Theorem 3.1. Thus(1) holds.

For (2), first observe that f ′ is definable in (R, <,+, ·, f). Let (F,<,⊕,⊗) beconstructed from f as in the proof of Theorem 3.1. Since F is nowhere dense,we obtain by Lemma 3.4 a (R, <,+, ·, f)-definable set Z ⊆ F such that the or-dered field isomorphism expands to an isomorphism between (F,<,⊕,⊗, Z) and


(R, <,+, ·,Z). An inspection of the proof of Theorem 3.1 shows that the iso-morphism (F,<,⊕,⊗, Z) → (R, <,+, ·,Z) given by x 7→ h′x(0) is (R, <,+, ·, f)-definable. It follows that (R, <,+, ·, f) is type C. �

The following corollary shows in particular that if R is type B and does not de-fine isomorphic copy of the standard model of second order arthimetic, then everydefinable C1 function f : I → R is affine. This is a special case of Question 1.4.

Corollary 3.7. Let K be a subfield of R such that R defines a dense ω-orderablesubset of K. Then

(1) if R is not type C, then every definable C2 function f : I → R is affinewith slope in K.

(2) if R does not define an isomorphic copy of (P(N),N,∈,+, ·), then everydefinable C1 function f : I → R is affine with slope in K.

Proof. First observe that if R does not define an isomorphic copy of (P(N),N,∈,+, ·), then R is not type C. Since R defines a dense ω-orderable set, it has tobe type B. Therefore R is not field-type by Fact 1.3. Thus by Theorem 3.1 everydefinable C2 function f : I → R is affine. Moreover, if in addition R does notdefine an isomorphic copy of (P(N),N,∈,+, ·), then every definable C1 functionf : I → R is affine by Theorem 3.1. It is left to show that the slope of a definableaffine function f : I → R is in K. Towards a contradiction suppose there is sucha function with slope α /∈ K. Its definability immediately implies definability ofx 7→ αx on [0, 1]. Let D be a dense ω-orderable subset of K. Note that αD is alsodense in R. Observe D − D ⊆ K and α(D − D) ⊆ αK. Since α /∈ K, we haveαK ∩K = {0}. This yields

(D −D) ∩ (αD − αD) = {0}.Thus R is type C by Fact 2.11. A contradiction. �

Let C be the middle-thirds Cantor set, or one of the generalized Cantor sets dis-cussed in [4]. It is observed in the proof of [18, Corollary 3.10] and the introductionof [4] that (R, <,+, C) defines a dense ω-orderable subset of Q. Since (R, <,+, C)has a decidable theory, it can not define an isomorphic copy of (P(N),N,∈,+, ·).Therefore Corollary 3.7 shows that if f : I → R is C1 and (R, <,+, C, f) does notdefine an isomorphic copy of (P(N),N,∈,+, ·), then f is affine with rational slope.

4. The one-variable case of Theorem B

In this section we prove Theorem B for one-variable functions f : I → R.

Theorem 4.1. A continuous definable function f : I → R is generically Ck forany k ≥ 1.

Theorem 4.1 and Theorem 2.4 together show that if f : U → R is DΣ, where U ⊆ Ris open and definable, then f is generically Ck.

The reader will find it helpful to have copies of [18, 26] handy, as we repeatedlymake use of results from these papers. We need to include a remark about thework in [26]. By [26, Theorem B], an expansion of (R, <,+) that does not definean isomorphic copy of (P(N),N,∈,+1) is type A. In Sections 3 and 4 of [26] allresults were stated for expansions that do not define (P(N),N,∈,+1). However, the

A TETRACHOTOMY 15

proofs only made use of the fact that such structures are type A. This should havebeen made clear, but the authors did not anticipate the relevance of the weakerassumption.

4.1. Prerequisites. Throughout this subsection R is type A. Before diving intothe proof we establish a few basic facts for later use.

Lemma 4.2. Let X ⊆ I ×R>0 be DΣ such that Xx is finite for every x ∈ I. Thenthere is a nonempty open J ⊆ I and ε > 0 such that J × [0, ε] is disjoint from X.

Proof. Let π : I×R>0 → R be the projection onto the first coordinate. By Fact 2.1π(X) is DΣ. Therefore π(X) either has interior or is nowhere dense by SBCT. Ifπ(X) is nowhere dense, then there is an open subinterval J ⊆ I that is disjointfrom π(X). For this subinterval J we get that J × R≥0 is disjoint from X.

Now suppose that π(X) has interior. Let I ′ be an open subinterval of I containedin π(X). After replacing I with I ′ and X with X ∩ [I ′ × R], we may suppose thatπ(X) = I. Let {Bs,t : s, t ∈ R>0} be a definable family of compact sets witnessingthat X is DΣ. Let

Cs,t = π(X \Bs,t) and Ds,t = I \ Cs,t for all s, t > 0.

As π(X) = I, we have x ∈ Ds,t if and only if Xx ⊆ (Bs,t)x. Since each Xx is finite,every x ∈ I is contained in some Ds,t. Thus

⋃s,tDs,t = I. By the classical Baire

Category Theorem there are s, t ∈ R>0 such that Ds,t is somewhere dense. Fixsuch s and t. Then X \ Bs,t is the intersection of a DΣ set by an open set andis thus DΣ. Hence Cs,t is DΣ as well. Because Ds,t is somewhere dense and thecomplement of a DΣ set, Ds,t has interior by SBCT. Let J be an open subintervalwhose closure is contained in the interior of Ds,t. Then

X ∩ [Cl(J)× R>0] = Bs,t ∩ [Cl(J)× R>0] .

As Cl(J)× {0} and Bs,t are disjoint compact subsets of R2, there is an ε > 0 suchthat no point in Bs,t lies within distance ε of any point in Cl(J) × {0}. For suchan ε the set J × [0, ε] is disjoint from Bs,t, and thus disjoint from X. �

Definition 4.3. A subset D of R>0 is a sequence set if it is bounded and discretewith closure D ∪ {0}.

It is easy to see that (D,>) has order type ω when D is sequence set. By [26, Lemma3.2] our expansion R either defines a sequence set or every bounded nowhere densedefinable subset of R is finite.

Lemma 4.4. Let D be a definable sequence set and let X ⊆ D × R be definablesuch that Xd is nowhere dense for each d ∈ D. Then

⋃d∈DXd is nowhere dense.

Proof. We first write⋃d∈DXd as an increasing union. Set

Y := {(d, x) ∈ D × R : ∃e ∈ D e ≥ d ∧ (e, x) ∈ X}.

Observe that⋃d∈DXd =

⋃d∈D Yd. Because Yd ⊆ Ye when d ≥ e, the family

{Yd : d ∈ D} is increasing. As (D,>) has order type ω, the set {e ∈ D : d ≤ e}is finite for every d ∈ D. Therefore Yd is a finite union of nowhere dense sets,and hence nowhere dense for every d ∈ D. By [26, Lemma 3.3] the set

⋃d∈D Yd is

nowhere dense. It follows directly that⋃d∈DXd is nowhere dense. �


4.2. Proof of Theorem 4.1. Throughout this subsection R is type A. Let I =(a, b), f : I → R, and h = (h1, . . . , hk) ∈ Rk. We define the generalized k-thdifference of f as follows:

∆0f(x) := f(x),

and for k ≥ 1

∆khf(x) := ∆k−1

(h1,...,hk−1)f(x+ hk)−∆k−1(h1,...,hk−1)f(x).

Observe that ∆k(l1,l2)f(x) = ∆1

l2∆k−1l1

f(x) whenever l1 ∈ Rk−1 and l2 ∈ R. Note

that for given h, the function ∆khf is defined on the interval (a, b− k‖h‖).

In the proof of the o-minimal case of Theorem 4.1 in [31], one only has to considerthe usual k-th difference (that is the case when h1 = · · · = hn). Our proof ofTheorem 4.1 however depends crucially on allowing the hi to differ. The reason forthis difference between the two proofs is that the o-minimal frontier inequality isapplied in [31], and this inequality doesn’t hold for type A expansions in general.

Let J be an open subinterval of I and k ∈ N. A tuple (u, x) ∈ Rk≥0 × J is (J, k)-

suitable if x+ k‖u‖ ∈ J . We denote the set of such pairs by SJ,k. Note that SJ,kis open and ∆k

hf(x) is defined for each (h, x) ∈ SJ,k.

If I = (a, b) and (x, h) ∈ I × R>0, then ((h, . . . , h), x) ∈ SI,k if and only ifa < x < x+ kh < b.

The following fact about generalized k-th differences follows easily by applyinginduction to k. We leave the details to the reader.

Lemma 4.5. Let k ∈ N, h = (h1, h2) ∈ R×Rk−1 and x ∈ R such that (h, x) ∈ SI,k.Let f, g : I → R be two functions. Then

(1) ∆k(h1,h2)f(x) = ∆k−1

h2∆1h1f(x), and

(2) ∆kh(f + g)(x) = ∆k

hf(x) + ∆khg(x).

Definition 4.6. We say Hfk holds on J if either

• ∆k(h,...,h)f(x) ≥ 0 for all (x, h) ∈ J × R>0 with ((h, . . . , h), x) ∈ SJ,k or

• ∆k(h,...,h)f(x) ≤ 0 for all (x, h) ∈ J × R>0 with ((h, . . . , h), x) ∈ SJ,k.

As in [31] our proof of Theorem 4.1 is based on the following theorem of Boas andWidder.

Fact 4.7 ([7, Theorem]). Let f : I → R be continuous and k ≥ 2. If Hfk holds on

I, then f (k−2) exists and is continuous on I.

Before proving Theorem 4.1 we establish Lemma 4.8. Loosely speaking, it statesthat in order to show that the generalized k-th difference is non-negative on a givenset, it is enough to prove that the k-th difference is non-negative on a subset whoseprojection onto the first coordinate is a sequence set.

Lemma 4.8. Let f : J → R be a continuous definable function and D be a definablesequence set. If ∆k

(d,h)f(x) ≥ 0 for all ((d, h), x) ∈ SJ,k ∩ (D × Rk−1) × J , then

∆kuf(x) ≥ 0 for all (u, x) ∈ SJ,k.

A TETRACHOTOMY 17

Proof. By continuity of f and openness of SJ,k, it is enough to show that {(u, x) ∈SJ,k : ∆k

uf(x) ≥ 0} is dense in SJ,k. Let U ⊆ SJ,k be open. Let (u1, u2, x) ∈ U ,where u1 ∈ R and u2 ∈ Rk−1. Because D is a sequence set, there are n ∈ N andd1, . . . , dn ∈ D such that (

∑ni=1 di, u2, x) ∈ U . It is left to show the following claim:

For every j ∈ {1, . . . , n}, (∑ji=1 di, u2, x) ∈ SJ,k and ∆k

(∑j

i=1 di,u2)f(x) ≥ 0.

First observe that since∑ji=1 di <

∑ni=1 di and (

∑ni=1 di, u2, x) ∈ SJ,k, we have

(∑ji=1 di, u2, x) ∈ SJ,k. We now show the second statement of the claim by applying

induction to j. For j = 1, ∆k(d1,u2)f(x) ≥ 0 by our assumptions on D. So now let

j > 1 and suppose ∆k(∑j−1

i=1 di,u2)f(x) ≥ 0. Since (

∑ji=1 di, u2, x) ∈ SJ,k, it follows

immediately that (dj , u2, x+∑j−1i=1 di) ∈ SJ,k. Thus ∆k

(dj ,u2)f(x+∑j−1i=1 di) ≥ 0 by

our assumption on D. Applying Lemma 4.5 and using our induction hypothesis weobtain

∆k(∑j

i=1 di,u2)f(x) = ∆k−1

u2∆1∑j

i=1 dif(x) = ∆k−1

u2

(f

(x+

j∑i=1

di

)− f(x)

)

=∆k−1u2

(f

(x+

j∑i=1

d1

)− f

(x+

j−1∑i=1

di

)+ f

(x+

j−1∑i=1

di

)− f(x)

)

=∆k−1u2

(∆1djf

(x+

j−1∑i=1

di

)+ ∆1∑j−1

i=1 dif(x)

)

=∆k−1u2

∆1djf

(x+

j−1∑i=1

di

)+ ∆k−1

u2∆1∑j−1

i=1 dif(x)

=∆k(dj ,u2)f

(x+

j−1∑i=1

di

)+ ∆k

(∑j−1

i=1 di,u2)f(x) ≥ 0.

�

We are now ready to prove the following stronger version of Theorem 4.1. It statesthat the open dense set on which the continuous function f is Ck, can be defineduniformly in the parameters that define f . We will use the stronger form to provethe multivariable case of Theorem B.

Theorem 4.9. Let Z ⊆ Rn be definable, let (Iz)z∈Z be a definable family of boundedopen intervals, and let (fz : Iz → R)z∈Z be a definable family of continuous func-tions. Then there is a definable family (Uz)z∈Z of open dense subsets of Iz suchthat fz is Ck on Uz for every z ∈ Z.

Proof. For z ∈ Z, let az, bz ∈ R be such that Iz = (az, bz). We show that for everyk ∈ N there is a definable family (Uz) of open dense subsets of Iz such that fz isCk on Uz for each z ∈ Z.

We first treat the case when R defines a sequence set D. By Fact 4.7 it is enough toshow that for every k ∈ N there is a definable family (Uz)z∈Z of open dense subsetsof Iz such that for every z ∈ Z and for every connected component J of Uz

• ∆khfz(x) ≥ 0 for all (h, x) ∈ SJ,k, or

• ∆khfz(x) ≤ 0 for all (h, x) ∈ SJ,k.


We proceed by induction on k. The case k = 1 follows easily from Fact 2.5.

Let k > 1. Observe that for every z ∈ Z and d ∈ D, ∆kd,hfz = ∆k−1

h ∆1dfz and that

∆1dfz is defined on the interval (az, bz − d). By the induction hypothesis there is a

definable family (Uz,d)(z,d)∈Z×D of dense open subsets of (az, bz − d) such that foreach connected component J of Uz,d, either

• ∆k−1h ∆1

dfz(x) ≥ 0 for all (h, x) ∈ SJ,k−1, or

• ∆k−1h ∆1

dfz(x) ≤ 0 for all (h, x) ∈ SJ,k−1.

For (z, d) ∈ Z ×D set

Xz,d :=(

(a, b− d) \ Uz,d)∪ {b− d}.

By Lemma 4.4 the set⋃d∈DXz,d is nowhere dense for each z ∈ Z. Set

Uz := Iz \ Cl

(⋃d∈D

Xz,d

).

Observe that (Uz)z∈Z is a definable family of dense open subsets of Iz.

Let z ∈ Z and let J be a connected component of Uz. Then for each d ∈ D, either

(i) (a, b− d) ∩ J = ∅ or(ii) J ⊆ (a, b− d) and one of the following is true:

(a) ∆k−1h ∆1

dfz(x) ≥ 0 for all (h, x) ∈ SJ,k−1, or

(b) ∆k−1h ∆1

dfz(x) ≤ 0 for all (h, x) ∈ SJ,k−1.

Since D is a sequence set, there are infinitely many d ∈ D for which (ii) holds.Denote the set all such d ∈ D by D′. Let

D′′ := {d ∈ D′ : ∆k−1h ∆1

df(x) ≥ 0 for all (h, x) ∈ SJ,k−1}.

Then either D′ \ D′′ is infinite or D′′ is infinite. Suppose D′′ is infinite. Wenow want to show that ∆k

ufz(x) ≥ 0 for all (u, x) ∈ SJ,k. By Lemma 4.8 it isenough to show that ∆k

d,hfz(x) ≥ 0 for all ((d, h), x) ∈ SJ,k ∩ (D′′×Rk−1)×J . Let

(d, h, x) ∈ SJ,k∩(D′′×Rk−1)×J . By definition of SJ,k, we get that x+k‖(d, h)‖ ∈ J .Thus x+ (k − 1)‖h‖ ∈ J and hence (h, x) ∈ SJ,k−1. Since d ∈ D′′, we get

∆kd,hfz(x) = ∆k−1

h ∆1dfz(x) ≥ 0.

The case when D′ \D′′ is infinite may be handled similarly.

We now suppose that R does not define a sequence set. By [26, Lemma 3.2] everybounded nowhere dense definable subset of R is finite. Set

Sz := {(x, h) ∈ Iz × R>0 : (h, . . . , h, x) ∈ SIz,k+2}.

and

V1,z := {(x, h) ∈ Sz : ∆k+2(h,...,h)fz(x) ≥ 0}

V2,z := {(x, h) ∈ Sz : ∆k+2(h,...,h)fz(x) ≤ 0}.

Observe that Sz is open and both V1,z and V2,z are closed in Sz. Let

Wz := Sz \ (IntV1,z ∪ IntV2,z) .

A TETRACHOTOMY 19

Then Wz is DΣ for each z ∈ Z. Since Wz ⊆ (V1,z \ IntV1,z) ∪ (V2,z \ IntV2,z), wehave that Wz is nowhere dense and therefore has no interior. It follows immediatelyfrom [18, Fact 2.9(1) & Proposition 5.7] that dimWz ≤ 1. Let π : R×R>0 → R bethe coordinate projection onto the first coordinate. Consider

Yz := {x ∈ I : dim(Wz)x = 1}.

By [18, Fact 2.14(2)] the set Yz is DΣ for each z ∈ Z. By [18, Theorem 3], we getthat dimYz = 0. Hence Yz is nowhere dense. Now let Uz be the complement ofCl(Yz) in Iz. From the definition of Yz we obtain that dim(Wz)x = 0 for all x ∈ Uz.In particular, each (Wz)x is nowhere dense and hence finite. Consider

Vz := {x ∈ Uz : ∀δ, ε > 0 (x− δ, x+ δ)× (0, ε) ∩Wz 6= ∅}.

We will show that Vz is nowhere dense for each z ∈ Z. Suppose J is an open subin-terval of Iz in which V is dense. Observe that (J × R>0) ∩Wz is DΣ. ApplyingLemma 4.2 to this set we get a subinterval J ′ ⊆ J and an ε > 0 such that J ′× (0, ε)is disjoint from Wz. This contradicts the density of V in J . Thus V is nowheredense.

Let U ′z be the complement of Cl(Vz). It is left to show that for the each z ∈ Z,the function fz is Ck on U ′z. Let x ∈ U ′. As x /∈ Vz, there are δ, ε > 0 such that(x−δ, x+δ)×(0, ε)∩Wz = ∅. It follows from connectedness that (x−δ, x+δ)×(0, ε)is contained in Int(V1,z) or Int(V2,z). If necessary decrease δ so that 2δ < (k + 2)ε.

Then it is easy to check that Hfzk+2 holds on (x− δ, x+ δ). By Fact 4.7 the function

fz is Ck on (x− δ, x+ δ). �

5. Proof of Theorem A

In this section we will prove Theorem A. For the convenience of the reader wefirst recall the statement of the theorem.

Theorem A. Suppose R is type A. The following are equivalent:

(1) R is field-type,(2) there is a DΣ field (X,⊕,⊗) with dimX > 0,(3) there is a DΣ family (Ax)x∈B of subsets of Rn such that dimB ≥ 2, each Ax

is one-dimensional, and Ax ∩Ay is zero-dimensional for distinct x, y ∈ B,(4) there is a definable open U ⊆ Rm and a DΣ function f : U → Rn that is

nowhere locally affine,(5) there is a DΣ function f : I → R that is nowhere locally affine.

We complete the proof in several steps. We first establish that (4) implies (5). Infact, we prove the following more general result that does not require the assumptionthat R is field-type.

Lemma 5.1. If every continuous definable f : I → R is generically locally affine,then every continuous definable f : U → Rm, where U ⊆ Rk is open, is genericallylocally affine.

Now Lemma 5.1 and Theorem 2.4 give that (4) implies (5). For the proof Lemma 5.1we need the following basic fact from analysis.


Fact 5.2. A continuous f : J → R is affine if and only if

f(x) + f(y)

2= f

(x+ y

2

)for all x, y ∈ J.

We also need a selection theorem for locally closed sets. A set X ⊆ Rn is locallyclosed if for every point x ∈ X there is an open set U ⊆ Rn containing x such thatX ∩ U is closed in U . Given C ⊆ Rn and p ∈ Rn, we let

d(p, C) := inf{‖p− q‖ : q ∈ C}.We also let Bn(q, r) be the open ball in Rn with center q and radius r > 0.

Lemma 5.3. Let A ⊆ Rm × Rn be definable such that Ap is locally closed for allp ∈ Rm. Let π be the coordinate projection Rm × Rn → Rm. Then there is adefinable function g : π(A)→ Rn such that (p, g(p)) ∈ A for all p ∈ π(A).

Proof. We first reduce to the case when Ap is bounded for every p ∈ Rm. Letf : π(A)→ R be given by

f(p) = inf{r ∈ R>0 : Bn(0, r) ∩Ap 6= ∅}+ 1.

Then {Bn(0, f(p)) ∩ Ap : p ∈ π(A)} is a definable family of nonempty boundedlocally closed sets. So we may assume that each Ap is bounded. For each p ∈ π(A)let Wp be the union of all open boxes B of diameter at most one in Rn such thatB ∩Ap is closed in B. Then {Wp : p ∈ π(A)} is a definable family of bounded opensets such that Ap ⊆ Wp and Ap is closed in Wp for all p ∈ π(A). Let C be the setof (p, q) ∈ Rm × Rn such that (p, q) ∈ A and

d(q,Rn \Wp) = max{d(x,Rn \Wp) : x ∈ Ap}.It is easy to see that C is definable and each Cp is nonempty and compact. Letg : π(A)→ Rn be the function that maps p ∈ π(A) to the lexicographically minimalelement of Cp. It is easy to check that g is definable and that (p, g(p)) ∈ A for allp ∈ π(A). �

Proof of Lemma 5.1. Let

f(x) = (f1(x), . . . , fm(x)) for all x ∈ Rk.Suppose that for each 1 ≤ i ≤ m there is an open dense definable subset Ui of U onwhich fi is affine. Then f is affine on U1∩ . . .∩Um. Thus without loss of generality,we can assume that m = 1.

We apply induction on k. The base case k = 1 holds by assumption. Let B := I1×. . .×Ik be a box contained in U . We show that there is a nonempty box contained inB on which f is affine. The argument goes through uniformly and therefore showsthat f is locally affine on a dense open subset of U . Let B′ = I1× . . .×Ik−1 and letπ : B → B′ be the projection away from the last coordinate. Define fx(t) := f(x, t)for all (x, t) ∈ B′×Ik. For each δ > 0 we define Eδ to be the set of all (z, t) ∈ B′×Iksuch that (t − δ, t + δ) is a subset of Ik on which fz is affine. Note that Eδ ⊆ Eδ′

when δ′ < δ. By Fact 5.2 the restriction fz to (t− δ, t+ δ) is affine if and only if

fz(x) + fz(y)

2= fz

(x+ y

2

)for all x, y ∈ (t− δ, t+ δ).

Continuity of f therefore implies each Eδ is closed. Let E be⋃δ>0Eδ. Then E

is Fσ and E is the set of (z, t) ∈ B′ × Ik such that fz is locally affine at t. By

A TETRACHOTOMY 21

our assumption E contains a dense open subset of {z} × Ik for all z ∈ B′. Anapplication of Fact 2.9 shows that E has interior. After shrinking B if necessary,we can assume that fz is affine on Ik for all z ∈ B′.

Let α, β : B′ → R be such that fx(t) = α(x)t+ β(x) for all x ∈ B′ and t ∈ Ik. Wefirst show that α is constant in x. Suppose not. Let q ∈ Q>0 and y, y′ ∈ Ik be suchthat y′ − y = q. Note that

α(z) = q−1[fz(y′)− fz(y)] for all z ∈ B′.

Thus α is definable and continuous. Since α is non-constant, the intermediate valuetheorem yields a nonempty open interval L contained in the range of α.

By continuity of α the set {z ∈ B′ : α(z) = s} is closed in B′ for all s ∈ L. ApplyingLemma 5.3 we obtain a definable g : L → B′ such that α(g(s)) = s for all s ∈ L.So fg(s) has slope s for all s ∈ L.

Let r ∈ Ik, r′ ∈ L, and δ > 0 be such that for all r − δ < t < r + δ we have t ∈ Ikand t+ (r′ − r) ∈ L. Let h : (r − δ, r + δ)→ R be given by

h(t) = fg(t+r′−r)(t)− fg(t+r′−r)(r).

Then for all r − δ < t < r + δ we have

h(t) = [(t+ r′ − r)(t) + β(g(t+ r′ − r))]− [(t+ r′ − r)(r) + β(g(t+ r′ − r))]

= t2 + t(r′ − 2r)− r′r + r2.

So h is definable and nowhere locally affine. Contradiction.

We have shown that α is constant on B′. Let a ∈ R be such that α(z) = afor all z ∈ B′. Therefore fx(t) = at + β(x) for all (x, t) ∈ B′ × Ik. Becauseβ(x) = fx(t)− ax for all (x, t) ∈ B′ × Ik, β is definable and continuous on B′. Byour induction hypothesis, β is affine on a box contained in B′. So after shrinking B′

if necessary we may assume that β is affine on B′. Thus f is affine on B′ × Ik. �

Proof of Theorem A. It is clear that (1) implies (2).

(2) ⇒ (3) : Let (X,⊕,⊗) be a DΣ-field such that dimX > 0 and X is a subset ofRn. Applying Corollary 2.7 we obtain a nonempty open interval I and a continuousdefinable injection g : I → X. For (a, a′) ∈ I2 we set

A(a,a′) := {(b, b′) ∈ I ×X : b′ =(g(a)⊗ g(b)

)⊕ g(a′)}.

Observe that each A(a,a′) is the graph of the function I → Rn given by

x 7→ [g(a)⊗ g(x)]⊕ g(a′).

This function is injective, because (X,⊕,⊗) is a field. By Theorem 2.8 we knowthat A(a,a′) is one-dimensional for every (a, a′) ∈ I2. Thus (A(a,a′))(a,a′)∈I2 is a

definable family of one-dimensional subsets of Rn+1 with dim I × I = 2.


Let (a1, a′1) and (a2, a

′2) be distinct elements of I2. It is left to show that A(a1,a′1) ∩

A(a2,a′2) is finite. Let (b, b′) ∈ A(a1,a′1) ∩A(a2,a′2). Then(g(a′1)⊗ g(b)

)⊕ g(a′1) =

(g(a2)⊗ g(b)

)⊕ g(a′2)

Since (X,⊕,⊗) is a field and g is injective, there is at most one such b. ThusA(a,a′) ∩A(b,b′) contains at most one element.

(3)⇒ (4) : Let (Ax)x∈B⊆Rl be a DΣ family of subsets of Rn such that dimB ≥ 2,dimAx = 1 for all x ∈ B, and dimAx ∩ Ay = 0 for distinct x, y ∈ B. Towards acontradiction, suppose that (4) fails.

Let A = {(x, y) ∈ B × Rn : y ∈ Ax}. We first show that we assume that B is anopen subset of R2. By Corollary 2.7 there is a nonempty definable open V ⊆ R2

and a continuous definable injection g : V → B. Set

A′ := {(p, q) ∈ V × Rn : (g(p), q) ∈ A}.

Then A′p = Ag(p) for all p ∈ V . Because A′ is the preimage of A under a continuousdefinable map, A′ is DΣ by Fact 2.1. After replacing A with A′ and B with V , wemay assume that B is an open subset of R2.

Let πk : Rn → R be the projection onto the k-th coordinate for k = 1, . . . , n. Aseach Ax is one-dimensional, Fact 2.6 shows that for all x ∈ B there is 1 ≤ k ≤ nsuch that πk(Ax) has interior. For k = 1, . . . , n, we set

Ak := {x ∈ B : dimπk(Ax) ≥ 1}.

It follows from [18, Fact 2.14] that Ak is DΣ for 1 ≤ k ≤ n. Since B =⋃nk=1Ak,

there is k such that Ak is somewhere dense in B. By SBCT there is k such thatAk has interior. Let us assume that A1 has interior. The case that Ak has interiorfor k with 2 ≤ k ≤ n, can be handled similarly. After replacing B with a definablenonempty open subset of A1 we may suppose that dimπ1(Ax) ≥ 1 for every x ∈ B.Let ρ : Rn+2 → R3 be the projection onto the first three coordinates. Note thatρ(A)x = π1(Ax) for all x ∈ R2. Thus dim ρ(A)x ≥ 1 for all x ∈ B. Since ρ(A) isDΣ, we know that dim ρ(A) = 3 by [18, Theorem F(3)]. Thus ρ(A) has interior byFact 2.6.

Let I, J, L ⊆ R be nonempty open intervals such that I×J×L ⊆ ρ(A). Thus for all(x, y, z) ∈ I × J × L there is an u ∈ Rn−1 such that (z, u) ∈ A(x,y). Applying DΣ-selection and replacing I, J, L with smaller nonempty open intervals if necessary,we obtain a continuous definable f : I×J×L→ R such that (z, f(x, y, z)) ∈ A(x,y)

for all (x, y, z) ∈ I × J × L. By our assumption that (4) fails, we can find opensubset U ⊆ I × J × L on which f is affine. Replacing I, J, L with even smallernonempty open intervals, we can assume that f is affine on I × J × L. Now fixlinear h1, h2, h3 : R→ Rn−1 and β ∈ Rn−1 such that

f(a1, a2, a3) = h1(a1) + h2(a2) + h3(a3) + β for all (a1, a2, a3) ∈ I × J × L.

Fix (u, v) ∈ I × J and u′ ∈ I such that u′ 6= u. Let v′ ∈ J satisfy

h1(u′) + h2(v′) = h1(u) + h2(v).

A TETRACHOTOMY 23

Then f(u, v, t) = f(u′, v′, t) for all t ∈ L. So {(t, f(u, v, t)) : t ∈ L} is a subset ofA(u,v) ∩A(u′,v′). We attain a contradiction as A(u,v) ∩A(u′,v′) is zero-dimensional.

Lemma 5.1 shows that (4) implies (5). It remains to establish that (5) implies(1). Suppose f : I → R is DΣ and nowhere locally affine. After applying the onevariable case of Theorem B and shrinking I if necessary, we may assume that f isC2. Because f is non-affine, R is field-type by Fact 1.1. �

6. Theorem B for multivariable functions

We assume that R is type A throughout this section. The goal is the proof ofTheorem B for multivariable functions. We begin with an outline of this proof. ByTheorem 2.4 it suffices to prove Theorem B for continuous definable functions. Letf : U → Rn be continuous and definable and U ⊆ Rm open. We first show thatgenerically all partial derivatives of f exist. However, in order to prove continuity ofthese derivatives, we have to invoke Theorem A. Indeed, by Theorem A, if R is notfield-type, then f is generically locally affine and hence generically C∞. Thereforeit suffices to treat the case when R is field-type. We further reduce this case tothe case that R is actually an expansion of (R, <,+, ·). In this situation, we canuse the definability of the partial derivatives of f to show that these derivatives arecontinuous almost everywhere.

Lemma 6.1. Let k ≥ 1, let U ⊆ Rn be an open definable set, let f : U → R bea continuous definable function, and let i be such that 1 ≤ i ≤ n. Then there is a

dense open definable set V ⊆ U such that ∂kf∂xi

(p) exists for all p ∈ V .

In the following proof we write ∆khf for ∆k

(h,...,h)f .

Proof. To simplify notation we suppose i = 1. Let W = I1 × . . .× In be a productof closed intervals with nonempty interior such that the closure of W is contained

in U . We show that ∂kf∂x1

exists and is continuous on a dense definable open subsetof W . Our argument goes through uniformly in W , so we obtain a dense definable

open subset of U on which ∂kf∂x1

exists and is continuous.

Let I1 = [a, b] and B = I2 × . . . × In. For x ∈ B we define fx : I1 → R to be the

function given by fx(t) := f(t, x) for t ∈ I1. Note that f(k)x (t) = ∂kf

∂x1(t, x) (if either

exists). As f is continuous, ∆k+2h fx(t) is a continuous function on the closed set

D := {(h, t, x) : h ≥ 0, (t, x) ∈W, t+ kh ≤ b}.Set

C1 := {(h, t, x) : (h, t, x) ∈ D,∆k+2h fx(t) ≥ 0}

and

C2 := {(h, t, x) : (h, t, x) ∈ D,∆k+2h fx(t) ≤ 0}.

Observe that C1 and C2 are closed definable sets. For i ∈ {1, 2}, set

Ei := {(δ, t, x) : δ ≥ 0, (t, x) ∈W and [0, δ]× [t− δ, t+ δ]× {x} ⊆ Ci}.Both E1 and E2 are closed definable sets. If δ > 0 and (δ, t, x) ∈ E1, then

∆(k+2)h fx(t) exists and is nonnegative on [t− δ, t+ δ]. Thus for such triples (δ, t, x),

the k-derivative f(k)x exists and is continuous on [t− δ, t+ δ] by Fact 4.7. Likewise,


if δ > 0 and (δ, t, x) ∈ E2, then f(k)x exists and is continuous on [t− δ, t+ δ].

Now for i ∈ {1, 2}, set

Fi := {(t, x) ∈W : ∃δ > 0 (δ, t, x) ∈ Ei}.

Note that F1 and F2 are DΣ. Set F := F1 ∪ F2. Note that F is DΣ, and that f(k)x

exists and is continuous in t for every (x, t) ∈ F . By the proof of Theorem 4.9, thereis a definable family (Ux)x∈B of open dense subsets of I such that Ux×{x} ⊆ F forevery x ∈ B. Therefore F contains an open dense subset of I × {x} for all x ∈ B.As F is Fσ, Fact 2.9 shows that F contains a dense open subset of W . Let V be

the interior of F . Then ∂kf∂x1

exists on V . �

We now establish a lemma allowing us to reduce certain questions about defin-able sets and functions in field-type expansions to questions about expansions of(R, <,+, ·). It is crucial for our proof of Theorem B, but we anticipate furtherapplications.

Lemma 6.2. Fix k ≥ 1. Suppose that R is field-type. Then there is an openinterval I, definable function ⊕,⊗ : I2 → I, an isomorphism τ : (I,<,⊕,⊗) →(R, <,+, ·), and J ⊆ I such that the restriction of τ to J is a Ck-diffeomorphismJ → τ(J).

Proof. By Theorem A, the expansionR defines a non-affine C2-function L→ R. Byinspection of the proof of Theorem 3.1 the reader can check that we can constructan open interval I, definable functions ⊕,⊗ : I2 → I, an isomorphism τ : (I,<,⊕,⊗) → (R, <,+, ·), J ⊆ I, and a definable, continuously differentiable functionf : J → R such that τ(x) = f ′(x) for all x ∈ J . After applying Theorem B andshrinking J if necessary, we may assume that f is Ck+1 on J . Thus τ is Ck on J .As τ = f ′ is strictly increasing, and f is C2, we also have τ ′(x) = f ′′(x) > 0 for allx ∈ J . By inverse function theorem the inverse τ−1 is Ck on τ(J). Therefore τ isa Ck-diffeomorphism J → τ(J). �

We now prove Theorem B for expansions of (R, <,+, ·).

Lemma 6.3. Suppose that R expands (R, <,+, ·). Let U ⊆ Rn be a definable openset, and let f : U → Rm be a continuous definable function. Then f is Ck almosteverywhere for all k ≥ 1.

Proof. Let f(x) = (f1(x), . . . , fm(x)) for all x ∈ Rn. Suppose that for i = 1, . . . , nthere is a dense definable open Vi ⊆ U on which fi is Ck. Then f is Ck on thedense definable open set V1 ∩ . . . ∩ Vm. We therefore suppose m = 1.

It suffices to show that for i = 1, . . . , n there is a dense open definable subset

Vi of U on which ∂kf∂xi

exists and is continuous. If this is true, then f is Ck onV1 ∩ . . .∩ Vn. Fix i with 1 ≤ i ≤ n. Applying Lemma 6.1 there is a dense definable

open set W ⊆ U such that ∂kf∂xi

(p) exists for all p ∈W . This is a definable function,

because R expands (R, <,+, ·). Furthermore ∂kf∂xi

is a pointwise limit of a sequence

of continuous functions W → R. An application of [18, Corollary 5.3] shows that∂kf∂xi

is continuous on a dense open subset V of W . Since the set of points at which∂kf∂xi

is continuous is definable, we may take V to be definable. �

A TETRACHOTOMY 25

Proof of Theorem B. The case when R is not field-type follows by Theorem A. Wetherefore suppose R is field-type. We show that any p ∈ U has a neighbourhoodW such that the restriction of f to W is Ck almost everywhere. This is enough asour proof is uniform in p. Fix p ∈ U for this purpose.

Applying Theorem 6.2 we obtain an I, definable ⊕,⊗ : I2 → I, an isomorphismτ : (I,<,⊕,⊗) → (R, <,+, ·), and J ⊆ I such that the restriction of τ to J is aCk-diffeomorphism J → τ(J).

Let τn : In → Rn be given by

τn(x1, . . . , xn) = (τ(x1), . . . , τ(xn)) for all (x1, . . . , xn) ∈ In.Then τn restricts to a Ck-diffeomorphism Jn → τ(Jn). Let S be the expan-sion of (R, <,+, ·) by all subsets of Rn of the form τn(X) for definable X ⊆ In.Then X ⊆ Rn is S-definable if and only if τ−1

n (X) ⊆ In is definable. A functiong : X → Rm is S-definable if and only if τ−1

m ◦ g ◦ τn : τ−1n (X)→ Rm is definable.

Let W := τn(Jn). After translating U if necessary we suppose p ∈ W . Aftershrinking J if necessary we suppose that W ⊆ U . Let g : W → Rm be given byg(x) = (τm ◦ f ◦ τ−1

n )(x) for all x ∈ W . Lemma 6.3 shows that g is Ck on a denseS-definable open subset V of W . As f = τ−1

m ◦ g ◦ τn, and τn and τ−1m are both

Ck this shows that f is Ck on τ−1n (U). Finally τ−1

n (U) is a dense open definablesubset of W . �

7. Linearity in type B expansions

In this section, we study the linearity of type B expansions. As noted in Question1.4 in the introduction, although type B expansions are not field-type, we do notknow whether every continuous function f : I → R definable in a type B expansionis generically locally affine. However, in the following two subsections, we are ableto obtain results indicating strong linearity of type B structures. We hope thatthese results might eventually lead to an affirmative answer of Question 1.4.

7.1. Repetitious functions. Let f : I → R. Recall that we say f is repetitiousif for every open subinterval J ⊆ I there are δ > 0, x, y ∈ J such that δ < y − xand

f(x+ ε)− f(x) = f(y + ε)− f(y) for all 0 ≤ ε < δ.

We now show Theorem E, which states that if R is type B and f is definable, thenf is repetitious. As similar result holds for linear type A structures. Observe thatif f is generically locally affine, then f is repetitious. Thus if R is type A and notfield type and f is definable, then f is repetitious by Theorem A.

Lemma 7.1. Suppose R is type B. Let D be an ω-orderable set that is dense in I,let f : I → R be definable and continuous, and let J ⊆ I be an open interval onwhich f is nonconstant. Then there are d1, d2, d3, d4 ∈ J ∩D such that

d1 6= d2, d3 6= d4 and d1 − d2 = f(d3)− f(d4).

Proof. Let J ⊆ I be an open subinterval on which f is not constant. Let a, b ∈ Rbe such that J = (a, b). The intermediate value theorem yields an open intervalJ ′ ⊆ f(J). Since D is dense in I, we have that f(D ∩ J) ∩ J ′ is dense in J ′. Leta′, b′ ∈ R be such that (a′, b′) = J ′. After decreasing b′ we assume b′ − a′ < b− a.


Then both (−a+D)∩ (0, b′ − a′) and −a′ +(f(D ∩ J)∩ J ′

)are dense ω-orderable

subsets of (0, b′ − a′). Applying Fact 2.11 to these two dense ω-orderable sets, weobtain d1, d2, d3, d4 ∈ J such that d1 6= d2, f(d3) 6= f(d4) and

(−a+ d1)− (−a+ d2) = −a′ + f(d3)−(− a′ + f(d4)

)It follows that d3 6= d4 and

d1 − d2 = (−a+ d1)− (−a+ d2) = −a′ + f(d3)−(− a′ + f(d4)

)= f(d3)− f(d4).

�

Proof of Theorem E. Suppose R is type B. Let f : I → R be continuous anddefinable. We need to show that f is repetitious. Let U be the set of p ∈ I atwhich f is locally constant. Note that the restriction of f to U is repetitious. LetV be the interior of I \ U . It suffices to show for every open interval L ⊆ V thatthe restriction of f to L is repetitious. We may therefore assume that there is noopen subinterval of I on which f is constant. Since R is type B, there is a denseω-orderable subset D of I. We declare

C := {(d1, d2, d3, d4) ∈ D4 : d1 6= d2, d3 6= d4}.

For d = (d1, d2, d3, d4) ∈ C, let Ad ⊆ R be the set of all t ∈ R such that

• t+ d3, t+ d4 ∈ I, and• d1 − d2 = f(t+ d3)− f(t+ d4).

For each d ∈ C, the set Ad is closed in I by continuity of f . Let s > 0 and J be anopen subinterval of I such that t+ J ⊆ I for all t ∈ (0, s). Let r ∈ (0, s). Considerthe function gr : I → R that maps c ∈ I to f(r+ c). Applying Lemma 7.1 to gr weobtain a d = (d1, d2, d3, d4) ∈ C such that:

d1 − d2 = gr(d3)− gr(d4) = f(r + d3)− f(r + d4).

Thus r ∈ Ad. Therefore (0, s) ⊆⋃d∈C Ad. By the Baire Category Theorem

Ad has interior for some d ∈ C. Fix such a d = (d1, d2, d3, d4) ∈ C and let J ′

be an open interval in the interior of Ad. Then the function J ′ → R given byt 7→ f(t+ d3)− f(t+ d4) is constant. The statement of the theorem follows. �

7.2. Weak Poles. In this section we give more restrictions on continuous functionsdefinable in type B expansions. Our results apply to a more general class of expan-sions, those that do not admit weak poles. A pole is a definable homeomorphismbetween a bounded and an unbounded interval.

Definition 7.2. A weak pole is a definable family {hd : d ∈ E} of continuousmaps hd : [0, d]→ R such that

(i) E ⊆ R>0 is closed in R>0 and (0, ε) ∩ E 6= ∅ for all ε > 0,(ii) there is a δ > 0 such that [0, δ] ⊆ hd([0, d]) for all d ∈ E.

Corollary 7.6 below shows that R admits a weak pole whenever it defines a pole.To our knowledge weak poles have not been studied before. We first prove TheoremC, which states that if R is type B, then R does not define a weak pole.

Proof of Theorem C. Towards a contradiction, supposeR defines a dense ω-orderableset (D,≺) and a weak pole {hd : d ∈ E}. Using Fact 2.10 we will show that R

A TETRACHOTOMY 27

is type C, contradicting our assumption that R is type B. After rescaling we mayassume that D is dense in [0, 1] and [0, 1] ⊆ hd([0, d]) for all d ∈ E. Set

Z := {(a, b) ∈ [0, 1]2 : a < b}.

Let λ : R>0 → E map x to the maximal element of (−∞, x] ∩ E. Let g : [0, 1] ×Z ×D → D map (c, a, b, d) to{

d, if c− a > λ(b− a);≺-minimal e ∈ D�d s.t. hλ(b−a)(c− a)− e is minimal, otherwise.

We will now show that g satisfies the assumptions of Fact 2.10. For this, let a, b ∈ Zand d, e ∈ D with e � d. As [0, 1] ⊆ hλ(b−a)([0, λ(b− a)]), there is z ∈ [0, λ(b− a)]such that hλ(b−a)(z) = e. Since D�d is finite and hλ(b−a) is continuous, there is anopen interval I around z such that for each y ∈ I, e is the only element in D�dsuch that hλ(b−a)(y)− e is minimal. Let c ∈ (a, b) be such that c−a = z. It followsimmediately from the argument above that g(x, a, b, d) = e for all x ∈ (c+I)∩(a, b).Thus (ii) of Fact 2.10 holds for our choice of g. Therefore R is type C. �

We now prove several results about continuous definable functions in expansionsthat do not admit weak poles. These results yield Theorem D.

Proposition 7.3. Suppose R does not admit a weak pole. Then every definablefamily {fx : x ∈ Rl} of linear functions [0, 1] → R has only finitely many distinctelements.

Proof. Let {fx : x ∈ Rl} be a definable family of linear functions [0, 1] → R thathas infinitely many distinct elements. After replacing each fx with |fx|, we mayassume that each fx takes nonnegative values. Let B = {fx(1) : x ∈ Rl} and letg : B × [0, 1] → R be given by g(λ, t) = fy(t) for any y ∈ Rl with fy(1) = λ. Notethat g is definable and g(λ, t) = λt for all (λ, t) ∈ B × [0, 1]. We declare

g(λ, t) = limλ′∈B,λ′→λ

g(λ′, t) for all (λ, t) ∈ Cl(B)× [0, 1].

By continuity we have that g(λ, t) = λt for all (λ, t) ∈ Cl(B)×[0, 1]. After replacingg by g and B by Cl(B), we may suppose that B is a closed and infinite subset ofR≥0. One of the following holds:

• B is unbounded.• B has an accumulation point.

First suppose that B is unbounded. Let {hd : d ∈ R>0} be the definable family offunctions hd : [0, d] → R given by declaring hd(t) = g(λ, t) where λ is the minimalelement of B such that g(λ, d) ≥ 1. Then hd(t) ≥ d−1t for all t ∈ [0, d]. It directlyfollows that {hd : d ∈ R>0} is a weak pole.

Now suppose (2) holds. Let µ be an accumulation point of B. We declare

ψ(λ, t) := |g(µ, t)− g(λ, t)| = |µ− λ|t for all λ ∈ B, t ∈ [0, 1].

Note that ψ is definable. Set

C := {|µ− λ| : λ ∈ B} = {ψ(λ, 1) : 0 ≤ λ ≤ 1}.

Observe that C is closed, definable, and contains arbitrarily small positive elementsas λ is an accumulation point of B. Let {hd : d ∈ C} be the definable family offunctions hd : [0, d] → R such that hd is the compositional inverse of t 7→ ψ(λ, t)


where λ ∈ B is such that d = |µ− λ| = ψ(λ, 1). Then hd satisfies hd(t) = d−1t. Itfollows that {hd : d ∈ C} is a weak pole. �

Proposition 7.4. Suppose R does not define a weak pole. Then every continuousdefinable f : I → R is uniformly continuous.

Proof. We first treat the case when m = 1. Suppose f : I → R is continuous,definable, and not uniformly continuous. We show that R defines a weak pole. Letδ > 0 be such that for all ε > 0 there are t, t′ ∈ I such that |f(t) − f(t′)| ≥ δ and|t− t′| ≤ ε. For every ε > 0 let

Aε := {t ∈ I : |f(t)− f(t′)| ≥ δ for some t ≤ t′ ≤ t+ ε}.

Note that each Aε is closed in I and nonempty. Let p be a fixed element of I. Letg0(ε) be the maximal element of Aε ∩ (∞, p] if Aε ∩ (∞, p] 6= ∅ and the minimalelement of Aε ∩ [p,∞) otherwise. Note that g0 : R>0 → I is definable. Let g1(ε) bethe least t′ ∈ [g0(ε), g0(ε) + ε] such that |f(g0(ε))− f(t′)| ≥ δ. Then g1 : R>0 → Iis definable and for all ε > 0:

0 < g1(ε)− g0(ε) ≤ ε and |f(g1(ε))− f(g0(ε))| ≥ δ.

We consider the definable family of functions hε : [0, g1(ε)− g0(ε)]→ R given by

hε(t) := |f(g0(ε) + t)− f(g0(ε))|.

Each hε is continuous. It follows from the intermediate value theorem that [0, δ] iscontained in the image of every hε. Thus {hε : ε ∈ R>0} is a weak pole. �

We leave the proof of Lemma 7.5, an easy consequence of the triangle inequality,to the reader.

Lemma 7.5. A uniformly continuous f : I → R on a bounded open interval Iis bounded. A uniformly continuous f : R>0 → R is bounded above by an affinefunction.

Proposition 7.4 and Lemma 7.5 together yield Corollary 7.6.

Corollary 7.6. Suppose R does not admit a weak pole. Then every continuousdefinable function on a bounded interval is bounded, and every continuous definablef : R>0 → R is bounded above by an affine function.

Proposition 7.7. Suppose R does not admit a weak pole. Suppose W is a boundeddefinable open subset of Rm. Then any continuous definable f : W → Rn isbounded.

Proof. Let f(x) = (f1(x), . . . , fn(x)) for all x ∈ W . It suffices to show that eachfi : W → R is bounded. So we suppose n = 1. Given t > 0 we let At be the set ofp ∈W such that ‖p− q‖ ≥ t for all q ∈ Rm \W . Note that each At is closed, as Wis bounded it follows that each At is compact. Let r > 0 be maximal such that Aris nonempty. Then At is nonempty for all 0 < t < r. Let g : (0, r]→ R be given by

g(t) = max{f(p) : p ∈ At}.

It is a routine analysis exercise to show that g is continuous. Corollary 7.6 showsthat g is bounded. It follows that f is bounded. �

A TETRACHOTOMY 29

8. Applications

8.1. Extensions of Fact 1.1. We give two extensions of Fact 1.1. The first is amultivariable version of Fact 1.1.

Theorem 8.1. Let U be a connected definable open subset of Rn, and let f : U →Rm be definable.

(1) If f is C2 and R is not field-type, then f is affine.(2) If f is C1 and R is neither field-type nor defines an isomorphic copy of

(P(N),N,∈,+, ·), then f is affine.

The proof is very similar to that of Lemma 5.1, so we will omit some details.

Proof. The proof of (2) follows by Theorem F and a similar argument as the proofof (1). So we only prove (1).Suppose that f : U → Rm is a definable C2-function and R is not field-type. Let

f(x) = (f1(x), . . . , fm(x)) for all x ∈ Rn.It suffices to show that fi is affine for i = 1, . . . ,m. Thus we reduce to the casethat m = 1.

As U is connected, it suffices to prove that f is affine on every open box containedin U . Hence, without loss of generality, we may assume that U = I1 × . . .× In is abox, where I1, . . . , In are open intervals. We proceed by induction on n. The basecase n = 1 is precisely Fact 1.1. Let U ′ = I1 × . . . × In−1 and let π : U → U ′ bethe projection away from the last coordinate. For x ∈ U ′, define fx : In → R byfx(t) = f(x, t) for all t ∈ In. Each fx is C2, so it follows that each fx is affine.

We show that f ′x(t) is constant on U ′. Suppose not. Following the proof ofLemma 5.1 we obtain a nonempty open interval J and r, r′ such that the func-tion h : J → R given by

h(t) = t2 + t(r′ − 2r)− r′r′ + r2

is definable. Then h is C2 and non-affine, contradiction.

Fix λ such that f ′x(t) = λ for all (x, t) ∈ U ′ × In. Let g : U ′ → R be such thatfx(t) = g(x) + λt for all (x, t) ∈ U . Since g(x) = fx(t) − λt for all (x, t) ∈ B,it follows that g is definable and C2. An application of induction shows that g isaffine. Thus f is affine as well. �

Recall that f : I → R is strictly convex if

f

(a+ b

2

)<f(a) + f(b)

2for all distinct a, b ∈ I.

A strictly convex function is continuous.

Theorem 8.2. Suppose R defines a strictly convex function. Then R is field-type.

Proof. Let f : I → R be a strictly convex definable function. Towards a contradic-tions, we suppose that R is not of field-type. Thus R is either type A or type B.By strict convexity we know that if x, y ∈ I, ε > 0 satisfy x+ ε < y and y + ε ∈ I,then

f(x+ ε)− f(x) < f(y + ε)− f(y).


Hence f is not reptitious. Therefore R can not be type B by Theorem E. However,a strictly convex function is also nowhere locally affine. Thus R can not be type Aeither by Theorem A. This is a contradiction. �

8.2. An application to descriptive set theory. We give an application to de-scriptive set theory. We assume that the reader is familiar with basic notions of thesubject (see Kechris [29] for an introduction). Consider the Polish space Ck([0, 1])of all Ck functions [0, 1] → R equipped with the topology induced by the semi-norms f 7→ maxt∈[0,1] |f (j)(t)| for 0 ≤ j ≤ k. Note that C0([0, 1]) is the space ofcontinuous functions [0, 1]→ R equipped with the topology of uniform convergence.We let C∞([0, 1]) be the space of smooth functions with the topology induced bythe semi-norms f 7→ maxt∈[0,1] |f (j)(t)| for j ∈ N. Grigoriev [21] and later Le Gal[32] constructed a comeager Z ⊆ C∞([0, 1]) such that (R, <,+, ·, f) is o-minimalfor all f ∈ Z. The corresponding result for Ck([0, 1]) fails.

Theorem 8.3. The set of all f ∈ Ck([0, 1]) such that (R, <,+, f) is type C, iscomeager in Ck([0, 1]) for any k ∈ N.

While it might not be surprising that expansions of (R, <,+) by a generic boundedcontinuous function are not model-theoretically well behaved, Theorem 8.3 actu-ally shows something stronger: a generic bounded continuous function defines allbounded continuous functions over (R, <,+). Loosely speaking, this means thatgiven two generic functions we can recover one from the other by using finitelymany boolean operations, cartesian products, and linear operations.

Sketch of proof of Theorem 8.3. It is well-known that the set of somewhere(k + 1)-differentiable functions in Ck([0, 1]) is meager, the case k = 1 being a clas-sical result of Banach [5]. Thus the set of all f ∈ Ck([0, 1]) such that (R, <,+, f) istype A, is meager by Theorem B. It therefore suffices to show that the collection ofall Ck functions [0, 1]→ R definable in type B expansions is meager. By TheoremE it is enough to prove that the set of reptitious f ∈ Ck([0, 1]) is meager. For eachn ≥ 1 let An be the set of functions f ∈ Ck([0, 1]) such that for some x, y ∈ [0, 1]

• 1n ≤ y − x, and

• f(x+ ε)− f(x) = f(y + ε)− f(y) for all 0 < ε ≤ 1n .

Note that every reptitious Ck-function [0, 1] → R is in some An. We show thateach An is nowhere dense. Let n ≥ 1. As An is a closed subset of Ck([0, 1]), weonly need to show that An has empty interior in Ck([0, 1]). For every f ∈ Ck([0, 1])and ε > 0, it is easy to construct a smooth g : [0, 1] → R such that g /∈ An and|f (j)(t)−g(j)(t)| < ε for all 0 ≤ j ≤ k and t ∈ [0, 1]. Thus An has empty interior. �

8.3. Applications to Automata Theory and automatic structures. We fin-ish with an application to automata theory. We first recall the terminology from[9]. Let r ∈ N≥2 and Σr = {0, . . . , r − 1}. Let x ∈ R. A base r expansion of x isan infinite Σr ∪ {?}-word ap · · · a0 ? a−1a−2 · · · such that

(1) z = − apr − 1

rp +

p−1∑i=−∞

airi

with ap ∈ {0, r− 1} and ap−1, ap−2, . . . ∈ Σr. We will call the ai’s the digits of thebase r expansion of x. The digit an is the digit in the position corresponding

A TETRACHOTOMY 31

to rn. We define Vr(x, u, k) to be the ternary predicate on R that holds when-ever there exists a base r expansion ap · · · a0 ? a−1a−2 · · · of x such that u = rn

for some n ∈ Z and an = k. We denote by Tr the expansion of (R, <,+) by Vr.By [4, Lemma 3.1] Tr defines a dense ω-orderable set, and by [9, Theorem 6] thetheory of Tr is decidable. Thus Tr is type B and does not interpret (P(N),N,∈,+, ·).

The connection to automata theory arises as follows. A set X ⊆ Rn isr-recognizable if there is a Buchi automaton A over the alphabet Σnr ∪{∗} whichrecognizes the set of all base-r encodings of elements of X. Such Buchi automataare also called real vector automata and were introduced in Boigelot, Bronneand Rasart [8]. By [9, Theorem 5] a subset of Rn is r-recognizable if and only if itis Tr-definable without parameters. From Corollary B we immediately obtain:

Corollary 8.4. Let f : I → R be C1 and non-affine. Then for every r ∈ N≥2, thegraph of f is not r-recognizable.

Block Gorman et al. [6] prove a generalization of Corollary 8.4: if f : I → R isdifferentiable and non-affine, then the graph of f is not r-recognizable. (This gen-eralization was attained after the proof of Corollary 8.4, but published first.) Oneadvantage of the more abstract proof of Corollary 8.4 is that it immediately gen-eralizes to other enumeration systems. The base r-numeration system above maybe replaced by other enumeration systems such as the β-numeration system usedin [11] (when β is a Pisot number) or the Ostrowski numeration system based ona quadratic irrational number used in [24]. These enumeration systems also giverise to type B structures with decidable theories. Thus analogues of Corollary 8.4also hold for these enumeration systems. Results similar to Corollary 8.4 have beenproven, for C2 functions, or for more restricted classes of automata, by Anashin [3],Konecny [30], and Muller [36].

As mentioned above Abu Zaid [1] has shown that (P(N),N,∈,+1) does not interpret(R, <,+, ·). So (P(N),N,∈,+1) cannot interpret an expansion of (R, <,+) of field-type. Applying this and Theorem 8.1 we obtain the following generalization ofCorollary 8.4.

Corollary 8.5. Let U be a connected definable open subset of Rn and let f : U →Rm be definable and C1. If R is interpretable in (P(N),N,∈,+1), then f is affine.In particular if R is ω-automatic with advice, then f is affine.

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Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801

Email address: [email protected]

URL: http://www.math.illinois.edu/~phierony

Department of Mathematics, University of California, Irvine

Email address: [email protected]

URL: https://www.math.uci.edu/~ewalsber

a tetrachotomy for expansions of the real

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