Volume 4 3

c M

CRM R PROCEEDING S &

LECTURE NOTE S Centre d e Recherche s Mathematique s Universite d e Montrea l

Additive Combinatorics Andrew Granvill e Melvyn B . Na thanso n Jozsef Solymos i Editors

The Centr e d e Recherche s Mathematique s (CRM ) of th e Universite d e Montrea l wa s create d i n 196 8 to promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Among it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , and publishing . Th e CR M is supporte d b y the Universit e de Montreal , th e Provinc e o f Quebec (FCAR) , and th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) of Montreal, whos e constituen t members ar e Concordi a University , McGil l University , the Universit e d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique . The CR M may b e reached o n th e Web a t www.crm.umontreal.ca .

American Mathematical Societ y Providence, Rhode Island US A

https://doi.org/10.1090/crmp/043

T h e product io n o f thi s volum e wa s suppor te d i n par t b y th e Fond s pou r l a Format io n de Chercheur s e t l 'Aid e a l a Recherch e (Fond s F C A R ) an d th e Na tu ra l Science s an d Engineering Researc h Counci l o f C a n a d a (NSERC) .

2000 Mathematics Subject Classification. P r i m a r y 11-02 ; Secondar y 05-02 , 42-02 , 11P70, 28D05 , 37A45 .

Library o f Congres s Cataloging-in-Publicatio n Dat a

CRM-Clay Schoo l o n Additiv e Combinatoric s (200 6 : Universit e d e Montreal ) Additive combinatoric s / Andre w Granville , Melvy n B . Nathanson , Jozse f Solymosi , editors .

p. cm . — (CR M proceeding s & lectur e notes , ISS N 1065-858 0 ; v. 43 ) Includes bibliographica l references . ISBN 978-0-8218-4351- 2 (alk . paper ) 1. Additiv e combinatorics—Congresses . 2 . Combinatoria l analysis—Congresses . I . Granville ,

Andrew. II . Nathanson , Melvy n B . (Melvy n Bernard) , 1944 - III . Solymosi , Jozsef , 1959 -IV. Title . V . Series .

QA164.C75 200 6 511'.5—dc22 200706083 4

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Contents

Preface v

An Introductio n t o Additiv e Combinatoric s Andrew Granville 1

Elementary Additiv e Combinatoric s Jozsef Solyrnosi 2 9

Many Additiv e Quadruple s Antal Balog 3 9

An Ol d Ne w Proo f o f Roth' s Theore m Endre Szemeredi 5 1

Bounds o n Exponentia l Sum s ove r Smal l Multiplicativ e Subgroup s Par Kuriberg 5 5

Montreal Note s o n Quadrati c Fourie r Analysi s Ben Green 6 9

Ergodic Method s i n Additiv e Combinatoric s Bryna Kra 10 3

The Ergodi c an d Combinatoria l Approache s t o Szemeredi' s Theore m Terence Tao 14 5

Cardinality Question s Abou t Sumset s Imre Z. Ruzsa 19 5

Open Problem s i n Additiv e Combinatoric s Ernest S. Croot III and Vsevolod F. Lev 20 7

Some Problem s Relate d t o Sum-Produc t Theorem s Mei-Chu Chang 23 5

Lattice Point s o n Circles , Square s i n Arithmeti c Progression s an d Sumset s o f Squares

Javier Cilleruelo and Andrew Granville 24 1

Problems i n Additiv e Numbe r Theory . I Melvyn B. Nathanson 26 3

Double an d Tripl e Sum s Modul o a Prim e Katalin Gyarmati, Sergei Konyagin and Imre Z. Ruzsa 27 1

iv C O N T E N T S

Additive Propertie s o f Produc t Set s i n Field s o f Prime Orde r A. A. Glibichuk and S. V. Konyagin 27 9

Many Set s Hav e mor e Sum s tha n Difference s Greg Martin and Kevin 0'Bryant 28 7

Davenport's Constan t fo r Group s o f the For m Z 3 © Z3 © Z3^ Gautami Bhowmik and Jan-Christoph Schlage-Puchta 30 7

Some Combinatoria l Grou p Invariant s an d Thei r Generalization s wit h Weight s S. D. Adhihari, R. Balasubramanian, and P. Rath 32 7

Preface

Andrew Granville , Melvy n B . Nathanson , an d Jozse f Solymos i

ABSTRACT. Pro m Marc h 30t h t o Apri l 5th , 2006 , a C R M - C l a y Schoo l o n Additive Combinatorics wa s hel d a t th e Universit e d e Montreal , followe d b y a worksho p fro m Apri l 6t h t o Apri l 12th , al l a s par t o f th e 2005-200 6 spe -cial yea r program , Analysis in Number Theory. Th e schoo l wa s attende d b y roughly on e hundre d participant s fro m sixtee n countrie s aroun d th e world , th e workshop b y fort y mor e people . Th e firs t par t o f thi s volum e contain s writte n versions o f mos t o f th e lecture s give n a t th e school ; th e secon d hal f submitte d contributions fro m th e speaker s a t th e workshop .

One of the most activ e areas in analysis today i s the rapidly emerging new topic of "additiv e combinatorics. " Buildin g o n Gowers ' us e o f the Freima n - Ruzsa theo -rem i n harmoni c analysi s (i n particular , hi s proo f o f Szemeredi' s theorem) , Gree n and Ta o famousl y prove d tha t ther e ar e arbitraril y lon g arithmeti c progression s o f primes, and Bourgain has given non-trivial estimates for hitherto untouchably shor t exponential sums . Thi s ne w subjec t bring s togethe r idea s fro m harmoni c analysis , ergodic theory , discret e geometry , combinatorics , grap h theory , grou p theory , prob -ability theor y an d numbe r theor y t o prov e som e extraordinar y results . Th e basi s of th e subjec t i s not to o difficult : i t ca n b e bes t describe d a s th e theor y o f addin g together set s of numbers; i n particular understandin g th e structur e o f the two orig-inal set s i f thei r su m i s small . Idea s fro m al l o f th e abov e area s com e i n whe n providing proof s o f ke y result s lik e th e Freima n - Ruzsa theorem , an d th e Balo g -Gowers - Szemered i lemma .

Because th e backgroun d i s s o broad , th e schoo l an d conferenc e attracte d a n eclectic mi x o f participants , bringin g differen t skill s an d perspectives . I t seem s evident tha t thi s combustible mixtur e wil l continue t o lea d t o exciting advance s fo r some tim e t o come .

The lecture s a t th e schoo l bega n wit h som e elementar y topics : Andrew Granville : The basics of additive combinatorics-, The Freiman - Ruzsa theorem; Uniform distribution and Roth's theorem. Joszef Solymosi : Combinatorial discrete geometry and additive combinatorics; a proof of Roth ;5 theorem. Antal Balog : The Balog - Szemeredi - Gowers Theorem.

We woul d lik e t o than k th e Centr e d e recherche s mathematiques , NSER C (Canada) , th e National Scienc e Foundatio n (USA) , th e Cla y Mathematic s Institut e (USA) , Dimati a (Czec h Republic) an d th e Universit e d e Montreal , fo r thei r generou s an d willin g suppor t o f ou r schoo l and workshop .

VI A. GRANVILL E E T AL .

The mai n lecturers , wh o eac h gav e a serie s o f talk s o n a dee p centra l theme , were Ben Green : Quadratic Fourier analysis. Bryna Kra : Ergodic methods in combinatorial number theory. Terry Tao : Combinatorial and ergodic techniques for proving Szemeredi-type the-orems. Van Vu : Structure of sumsets and applications.

The schoo l ende d wit h lecture s b y severa l o f the ke y figures i n th e earl y devel -opment o f wha t w e now cal l "additiv e combinatorics" : Imre Ruzsa : Pliinnecke and the others. Gregory Freiman : Inverse additive number theory: Results and problems. Endre Szemeredi : Another proof of Roth''s theorem.

The main lecture series of the school was extraordinarily successful . Al l of these lecturers mad e thei r talk s highl y accessible , whic h was reflected i n high attendanc e throughout th e meeting, and the buoyant atmosphere . W e believe that th e lecturer s have brought tha t attitud e t o the write-ups o f their contribution s herein ! W e would particularly lik e t o than k al l o f th e lecturer s fo r thei r super b talks , an d th e gen -erosity wit h whic h the y worke d wit h th e participants , helpin g the m t o understan d this challengin g material .

Many leadin g figures i n additiv e combinatorics , ergodi c theory , combinatorics , number theor y an d harmoni c analysi s arrive d fo r th e workshop , includin g Jea n Bourgain, Ti m Gowers , Mei-Ch u Chang , Ro n Graham , Trevo r Wooley , Michae l Lacey, Sanj u Velani , Serge i Konyagin , Jarosla v Nesetri l an d man y more . I n ad -dition, ther e wer e tw o beautifu l lecture s fo r th e genera l public : i n th e first week , Terry Ta o gave us further insight s int o Long arithmetic progressions in the primes; in the second week Manjul Bhargav a gave a beguiling introduction t o his work wit h Jonathan Hank e o n The representation of integers by quadratic forms. Ther e wer e a lo t o f announcements o f exciting ne w work i n the meetin g o n this ver y ho t topic , most notabl y perhap s Gree n an d Tao' s announcemen t tha t the y hav e a viable pla n to exten d thei r resul t o n primes i n arithmeti c progression s t o prov e a "wea k form " of the prime /c-tuplet s conjecture ; an d wor k by Helfgott , an d the n b y Bourgain an d Gambaud o n th e (non-abelian ) grou p generatio n problem .

This was just th e third worksho p on additive combinatorics (th e first was at th e American Institute of Mathematics i n September 2004 , the second a t th e Universit y of Bristo l i n Septembe r 2005) . Ove r th e nex t fe w year s there wil l be severa l majo r programs i n additiv e combinatoric s an d relate d areas , includin g durin g Fal l 200 7 at th e Institut e fo r Advance d Stud y i n Princeton , durin g Sprin g 200 8 a t th e Field s Institute i n Toronto , an d durin g Fal l 200 8 a t th e Mathematica l Science s Researc h Institute i n Berkeley , California .

The meetin g woul d no t hav e bee n possibl e withou t th e organizationa l skill s of Loui s Pelletie r an d hi s tea m fro m th e Centr e d e recherche s mathematiques , fo r which w e are ver y grateful .

PREFACE vn

This boo k i s spli t int o thre e parts : th e proceeding s fro m th e school , article s on ope n question s i n th e subject , an d ne w research . W e begi n wit h basi c articles , setting th e scene , by the organizers , alon g with a n explici t discussio n o f the Balog -Szemeredi-Gowers Theore m by Antal Balog, an old but unpublishe d proo f o f Roth' s theorem b y Szemeredi , an d a discussion o f Bourgain's bound s o n exponentia l sum s by Pa r Kurlberg . Andrew Granville : An Introduction to Additive Combinatorics. Joszef Solymosi : Elementary Additive Combinatorics. Antal Balog : Many Additive Quadruples. Endre Szemeredi : An Old New Proof of Roth's Theorem. Par Kurlberg : Bounds on Exponential Sums over Small Multiplicative Subgroups.

Next w e have thre e lectur e serie s describing som e o f the mos t excitin g idea s i n the curren t developmen t o f the subject . Ben Green : Quadratic Fourier analysis. Bryna Kra : Ergodic Methods in Combinatorial Number Theory. Terence Tao : The Ergodic and Combinatorial Approaches to Szemeredi's Theo-rem.

These ar e followe d b y severa l article s highlightin g ope n question s i n differen t aspects o f additive combinatorics . Imre Z . Ruzsa : Cardinality Questions about Sumsets. Ernest S . Croo t II I an d Sev a Lev : Open Problems in Additive Combinatorics. Mei-Chu Chang : Some Problems Related to Sum-Product Theorems. Javier Cilleruel o an d Andre w Granville : Lattice Points on Circles, Squares in Arithmetic Progressions and Sumsets of Squares. Melvyn B . Nathanson : Problems in Additive Number Theory. I .

This i s followe d b y severa l researc h article s correspondin g t o th e proceeding s of the workshop . Katalin Gyarmati , Serge i Konyagi n an d Imr e Z . Ruzsa : Double and Triple Sums Modulo a Prime. A. A . Glibichu k an d Serge i Konyagin : Additive Properties of Product Sets in Fields of Prime Order. Greg Marti n an d Kevi n O'Bryant : Many Sets Have More Sums Than Differ-ences. Gautami Bhowmi k an d Jan-Christop h Schlage-Puchta : Davenport's Con-stant for Groups of the Form Z 3 0 Z 3 © Z^d. S. D . Adhikari , R . Balasubramania n an d P . Rath : Some Combinatorial Group Invariants and Their Generalizations with Weights.

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