An Infinite Stringof Ants and Borel’sMethod ofSummabilityAVRAHAM FEINTUCH, AND BRUCE FRANCIS

IImagine some ants walking in single file, each pursuingthe one in front. By this we mean that at any givenmoment, the ant heads directly for the ant in front of him

and, if he wasn’t moving as well, would collide with him ina single unit of time.

If there arefinitelymany, there’s a first one.Nothavingoneto pursue, it could remain still; or it could head for a bush orsome other landmark; or it could pursue the last one, the oneat theendof the string. That third case is cyclic pursuit, and forsuitable velocity rules the ants converge to the average oftheir starting points. For example, if there are four ants, andthey begin at the four vertices of a square, they will all con-verge to the center of the square.

Butwhat if there are countably infinitelymany ants, so thatthere is no end to the string of ants in either direction? Whatwill happen then? Will they rendezvous, that is, converge to acommon point? That is the subject of this paper.

In cyclic pursuit, we start with a set q0; q1; . . .; qn of npoints in theplane, that represent the initial configurationof adynamical system of points that begin to move at time t = 0according to the law of motion

q0

iðtÞ ¼ qi�1ðtÞ � qiðtÞ; i ¼ 1; . . .;n

q0

0ðtÞ ¼ qnðtÞ � q0ðtÞ

for t [ 0. This simply means that point qi pursues pointqi-1, and point q0 pursues point qn. It is easy to show thatas t !1; each of the points converges to the averageq ¼ 1

nþ1

Pni¼0 qi:

We will be considering the analogous problem forbounded infinite sequences. More precisely, given a boun-ded sequence qnf g1n¼�1of complex numbers that is theinitial configuration at t = 0 of the linear dynamical system

q0

nðtÞ ¼ qn�1ðtÞ � qnðtÞ; n 2 Z;

for t [ 0, what happens as t !1?It is not difficult to see that if the initial configuration

sequence converges to a limit l as n tends to�1; then all thepoints will converge to l as t !1 (as we shall see, this can beattributed to E. Borel [2]). This of course generalizes the resultfor finitely many points: the limit of a convergent sequence isits Cesaro limit, which corresponds to the average in the finitecase. The first natural question is, what happens when thesequence has no limit but has a Cesaro limit? More generally,what can be said for an arbitrary bounded initial configura-tion? This is known as the serial pursuit and rendezvousproblem. This problem has been studied extensively in recentyears in the systems and control literature, both for finite andinfinite sets of points in the plane. In the case of an infinite setof points, we have a system of infinitely many linear equa-tions, with an initial configuration at time t = 0 serving as theinitial condition for the system.

It has usually been assumed that the sequence given in theinitial configuration belongs to the Hilbert sequence space‘2ðZÞ:We have explained elsewhere [5] why this assumptionis too restrictive and is not compatible with the physics of theproblem. It obliges all of our ants,whatever their preferences,

� 2012 Springer Science+Business Media, LLC, Volume 34, Number 2, 2012 15

DOI 10.1007/s00283-012-9286-z

to end up at the origin, which may be a nice place to visit, butwho wants to be stuck there for eternity? We will be morepermissive and specify only bounded initial configuration.We will see that the question of existence of a solution to therendezvous problem is in fact a question about Borel’smethod of summability for bounded sequences ([2], p. 15).This will lead us to a classical result of G. H. Hardy ([6],Theorem 149) that provides a partial solution to thequestionswe have raised.

The results presented here also appeared in [5] in a moregeneral context that deals with a more complex problem.

The Operator EquationWe consider, then, the linear system of equations

q0

nðtÞ ¼ qn�1ðtÞ � qnðtÞ; �1\n\1;

with initial condition

qnð0Þ ¼ qn;

where qnf g1�1 is a bounded sequence of complex numbers.We can of course write this in a more condensed form. Theinitial data become a single element of the space of boun-ded complex sequences, q ¼ . . .; q�2; q�1; q0; q1; q2; . . .ð Þ 2‘1ðZÞ; and the dynamical system concerns a variable pointq(t) in the same Banach sequence space. Introducing thefamiliar shift operator U on sequences, we write

q0 ðtÞ ¼ ðU � IÞqðtÞ

qð0Þ ¼ q:

The matrix representation of U - I with respect to the stan-dard co-ordinate vectors (which aren’t a basis in the Banachspace sense, but we will ignore that) is a doubly infinitematrix with -1 on its main diagonal and 1 on the diagonalbelow it, and with all other entries zero.

The norm of a vector in this space ‘1 is just the supremumof the sequence of absolute values of the coordinates of thevector, and U is the ‘‘bilateral right shift operator,’’ which

simply moves each coordinate one space to the right. Thus Uacting ona vectorwill not change its norm:U is an isometry. Itis not difficult to see that its spectrum is the unit circle,and every point in the spectrum is an eigenvalue with aone-dimensional eigenspace [5]. An eigenvector corre-sponding to the eigenvalue k = 1 is the constant sequence. . .; 1; 1; 1; . . .ð Þ; and this is a basis vector for Ker (U - I).

Since U is a bounded operator, the argument used for finite-dimensional linear systems can be applied verbatim to obtainthat at time t [ 0,q(t) is a vector in ‘1ðZÞ; and it is givenby thesolution to the operator differential equation, namely,

qðtÞ ¼ exp½ðU � IÞt�qð0Þ:

It follows immediately that the constant sequence. . .; 1; 1; 1; . . .ð Þ is a basis for the equilibrium subspace of the

system: If all the points qn are initially at the same spot, theywill not move, but otherwise they will. Also, the inducedoperator norm eðU�IÞt�

��� on ‘1 Zð Þ is equal to 1 for all t, and

thegroup {e(U-I)t} is thusuniformlyboundedas a functionof t.

By direct computation, the n-th coordinate of q(t) is

qnðtÞ ¼ e�tX1

k¼0

qn�kð0Þtk

k!;

and our question, what happens to the n-th point as t !1; becomes, given the sequence (n being fixed) {qn-k(0)},does limt!1 e�t

P1k¼0 qn�kð0Þ tk

k! exist? This is known as theproblem of the ‘‘Borel method of summability.’’ In sum-mability theory [2], if this limit exists and equals a, one saysthat the sequence {qn-k(0)} is Borel summable to a.

Our first result goes back to Borel using classical argu-ments. It is referred to as the regularity of the Borel method ofsummability, because it asserts that any convergent sequenceis Borel summable to the same limit. We present a ‘‘modern’’proof using uniform boundedness.

THEOREM 1 If limn!�1 qnð0Þ ¼ c; then q(t) converges in

‘1ðZÞ to the constant vector . . .; c; c; c; . . .ð Þ:

.........................................................................................................................................................

AU

TH

OR

S AVRAHAM FEINTUCH was born in the

UNRWA camp for Holocaust survivors in

Bergen Belsen, and he immigrated with his

family to Canada in 1949. He received his

Ph.D. in mathematics at the University ofToronto in 1972. Since then he has been at

the Ben Gurion University in Be’er Sheva,

working in operator theory, especially as

applied to system theory. He has published

a number of books and articles in this field,

and also on the Mishneh Torah of Maimo-

nides (1135–1204).

Department of Mathematics

Ben Gurion University

Be’er Sheva 84105Israel

e-mail: abie@math.bgu.ac.il

BRUCE FRANCIS was born in Toronto and

received all his university education at the

University of Toronto, culminating in a Ph.D.

in Electrical Engineering in 1975. After stints at

the University of California, Cambridge (Eng-

land), McGill, Waterloo, and Yale, he came in1984 to his present position in the ECE

Department at the University of Toronto. He

is the author of numerous papers on the

theory of automatic control systems, and he is

the winner of several prizes from the IEEE.

Department of Electrical and Computer

Engineering

University of Toronto

Toronto M5S 3G4

Canadae-mail: bruce.francis@utoronto.ca

16 THE MATHEMATICAL INTELLIGENCER

PROOF. By linearity we can assume that c = 0. Suppose

first that the initial configuration is given by the unit coordi-

nate vector d with d0 = 1 and dn = 0 for n = 0. Then

qðtÞk k1 ¼ eðU�IÞtd��

��1

¼ e�t eUtd��

��1

¼ e�t dþ tUdþ ðt2=2!ÞU 2dþ � � ���

��1

¼ e�t supk� 0

tk

k!

����

����

which approaches zero as t !1 (this is a nice calculusproblem for honour students).

Next, if q(0) is a finite linear combination of coordinatevectors, that is, of U kd

� �; it follows by linearity and the tri-

angle inequality that qðtÞk k1! 0 as t !1:Theclosed linearspan of finite linear combinations of the coordinate vectors isthe subspace c0 of vectors that converge to zero as n! �1:What remains is to extend our conclusion from the densesubspace where it is known, to all of c0. The requisite prop-erty is that the operators involved be uniformly bounded.Indeed, as we remarked previously, they are all of norm 1, sothe extension succeeds.

We will see later that there are bounded initial configu-rations for which the points do not rendezvous. However, ifthe n-th point converges to a given point as t !1; then thepoints following it will also converge to the same point, thatis, if limt!1 qnðtÞ ¼ a for n = n0, then this limit exists forevery n [ n0 and all the limits are equal. This is intuitivelyobvious, because ant number n0 + 1 pursues ant numbern0, ant number n0 + 2 pursues ant number n0 + 1, andso on. To prove the conclusion, we do best to definehn(t) = et qn(t), whence hn+1

0(t) = hn(t). Now if we know

limt!1 qnðtÞ ¼ a and wish to infer the same relation with nreplaced by n + 1, we calculate, using l’Hopital’s Rule,

limt!1

hnþ1ðtÞet

¼ limt!1

h0nþ1ðtÞet

¼ limt!1

hnðtÞet¼ a:

Ants apparently are well acquainted with l’Hopital. We don’tknow if anything comparable happens going the otherdirection along the procession. Does ant number n - 1 lookover his shoulder and see all his colleagues heading toward aparticular point and decide to join them there? We are notfamiliar enough with ant behaviour to answer this question.

We complete this section with an example from [4] of abounded sequence that is not Borel summable.

EXAMPLE 2 LetA be an infinite subset of the non-negative

integers. Under what conditions onAwill limt!1 e�tP

k2Atk

k!

exist? As shown in [4], Theorem 1, either of the following two

conditions is equivalent:

(1) Let Sn denote the number of heads that occur in n tossesof a coin. The condition is that limn!1 PrðSn 2 AÞ shouldexist.

(2) The condition is that

limn!1

1

�ffiffiffinp cardfk : k 2 A;n� k\nþ �

ffiffiffinpg

should exist for all �[ 0:

We want to exhibit failure of condition (2). It suffices to taken = m2 and � ¼ 1 and obtain an example where

1

mcðmÞ ¼ 1

mcardfk : k 2 A;m2� k\m2 þmg

does not converge as m!1: For this, it is enough to takeas A the union of the sets fm2;m2 þ 1; . . .;m2 þm� 1g form ¼ 3; 5; 7; . . .; for then c(m) = 0 for m even and c(m) =

m for m odd.

The Main ResultsWe saw (Theorem 1) that if the sequence given in the initialconfiguration of the system has a limit, then all the pointsconverge to this limit. We now consider the case where thissequence doesn’t converge, but there exists n0 such that thesequence qn0�kð0Þf g has Cesaro sum a.

THEOREM 3 Assume that for some n0 there exists a number

a such that

1

N þ 1

XN

k¼0

qn0�kð0Þ ¼ aþ o1ffiffiffiffiNp� �

:

Then limt!1 qnðtÞ ¼ a for every n.

PROOF. If the assumption holds for n0, then in fact it holds

for all n. This follows from the observation that for any n the

sequence qn�kð0Þf g differs from qn0�kð0Þf g only by finitely

many terms in the beginning of the sequence, and therefore

their Cesaro sums are identical with the same rate of con-

vergence.

By Theorem 149 of [6], the sequence qn0�kð0Þf g is Borel

summable to a. Thus this holds for all n, and for each

n; limt!1 qnðtÞ ¼ a:

Hardy ([6], 9.8 (iii)) shows that the sequence an =

(- 1)mm when n = m2 and an = 0 otherwise has Cesarosum zero but is not Borel summable. This sequence is ofcourse not bounded. Example 2, given previously from [4], isof course bounded and has Cesaro sum 1/2, but is not Borelsummable. Here is a cute example.

EXAMPLE 4 Let a be an irrational number that is not a

rational multiple of p. Kronecker’s density theorem says the

sequence einaf gn2Z is dense on the unit circle. We take this

sequence as the initial configuration of our system. It is ele-

mentary that for any n, {qn-k(0)} satisfies the condition of

Theorem 3 with Cesaro sum zero. Thus all the points in the

sequence will converge to the center of the circle as t ? ?.

This can easily be computed directly:

limt!1

qnðtÞ ¼ limt!1

e�tX1

k¼0

eiðn�kÞa tk

k!

¼ limt!1

e�teinaeðe�iaÞt

¼ limt!1

eðcos a�1Þteinae�it sin a

¼ 0:

Denote by a the constant sequence in ‘1ðZÞ with eachmember equal to a. Is it true under the assumption of

� 2012 Springer Science+Business Media, LLC, Volume 34, Number 2, 2012 17

Theorem 3 that qðtÞ � ak k1! 0 as t !1? Equivalently, is ittrue that

limt!1

supn

qnðtÞj j ¼ a?

Another question we don’t know the answer to (but maybe known to some experts in summability theory as well as toany serious sequence of ants) is, does there exist a boundedsequence that isBorel summablebutnotCesaro summable? Itwould seem from [1] that the answer should be yes. Thiswould mean that the sequence of points can converge to apoint that is not their center of gravity. What, then, is it? Can itbe given a physical meaning?

Our final result shows that although the points don’tnecessarily converge, their velocities converge to zero uni-formly as t !1: Wherever they are going, the ants willuniformly slow down to a crawl as time passes.

THEOREM 5 The ‘1-induced norm of (U - I)e(U-I)t con-

verges to 0 as t !1: Thus, for every qð0Þ 2 ‘1ðZÞ; q0 ðtÞconverges to zero in ‘1 Zð Þ as t !1; and, moreover, if q(0)

belongs to ðU � IÞ‘1; the range space of U - I acting on

‘1 Zð Þ; then q(t) converges to zero in ‘1 Zð Þ as t !1:

PROOF. Given qð0Þ 2 ‘1 Zð Þ; let r(t) = (U - I)e(U-I)tq(0).

The n-th component of r(t) is

rnðtÞ ¼ e�t �qnð0Þ þX1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

� �

qn�ðkþ1Þð0Þ" #

:

Thus

rðtÞk k1 ¼ e�t supn�qnð0Þ þ

X1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

� �

qn�ðkþ1Þð0Þ�����

�����

� e�t 1þX1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

����

����

" #

qð0Þk k1:

It therefore suffices (for both conclusions) to show that

limt!1

e�tX1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

����

���� ¼ 0:

It is elementary that for fixed t, the sequence tk

k!

n ohas a

maximum at some k0(t), and that this integer satisfies

k0(t) B t B k0(t) + 1. Also, for k B k0(t), the sequence is

increasing, and for k C k0(t), the sequence is decreasing.

Therefore,

X1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

����

����

¼Xk0ðtÞ�1

k¼0

tkþ1

ðk þ 1Þ!�tk

k!

� �

þX1

k¼k0ðtÞ

tk

k!� tkþ1

ðk þ 1Þ!

� �

¼ 2tk0ðtÞ

k0ðtÞ!� 1:

Thus

e�tX1

k¼0

tk

k!� tkþ1

ðk þ 1Þ!

����

����� 2e�t max

k

tk

k!;

which approaches zero as t !1: This completes theproof.

We make a number of remarks in order to view the lastresult from a broader perspective.

(1) The closure of Im (U - I) is an invariant subspace for thegroup eðU�IÞt� �

: The theorem says that if we restrict tothis subspace, the system is uniformly asymptoticallystable.

(2) The theorem can also be interpreted to mean thatalthough the system of points doesn’t necessarily con-verge, it does approach theone-dimensional equilibriumsubspace Ker (U - I) asymptotically.

(3) The subspace Im (U - I) is easily characterized. It is thesubspace of sequences in ‘1 Zð Þ whose sequence offinite partial sums is also in ‘1 Zð Þ:

Throughout this article we have considered the situationfor doubly infinite initial configurations. Is there an analoguein which the initial configuration is a one-sided infinitesequence? In this situation it is natural to have qi pursue qi+1.The appropriate operator is the ‘‘unilateral left shift’’ V on‘1ðZþÞ; defined by

V ðx0; x1; x2; . . .Þ ¼ ðx1; x2; x3; . . .Þ:

Then the n-th coordinate (for n C 0) of q(t) is

qnðtÞ ¼ e�tX1

k¼0

qnþkð0Þtk

k!;

and the analysis can proceed as mentioned previously.

REFERENCES

[1] A. Amir, Journal d’Analyse Mathematique 3 (1953), no. 1, 346–

381.

[2] J. Boos and F. P. Cass, Classical and Modern Methods in

Summability, Oxford, Oxford University Press, 2000.

[3] E. Borel, Lecons sur les series divergentes, 2nd Ed., Paris,

Gauthier-Villars, 1928.

[4] P. Diaconis and C. Stein, Some Tauberian Theorems Related to

Coin Tossing, Annals of Probability, 1978, 6, 3, 483–490.

[5] A. Feintuch and B. Francis, Infinite Chains of Kinematic Points,

preprint.

[6] G. H. Hardy, Divergent Series, Oxford, Clarendon Press, 1967.

18 THE MATHEMATICAL INTELLIGENCER