Additive problems in abelian groups

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  • Univeristy of WarsawFaculty of Mathematics, Informatics and Mechanics

    Karol Cwalina

    Additive problems in abeliangroups

    PhD dissertation

    Supervisordr hab. Tomasz SchoenWMI UAM, Pozna

    October, 2013

  • Authors declaration

    Aware of legal responsibility I hereby declare that I have written this dissertationmyself and all the contens of the dissertation have been obtainet by legal means.

    date Authors signature

    Supervisors declaration

    The dissertation is ready to be reviewed.

    date Supervisors signature

  • Abstract

    In this thesis we shall present some results concerning additive properties of finite sets inabelian groups. It will be of primary importance to us to consider the sumsets

    A+B = {a+ b : a A, b B}

    for subsets A,B of an abelian group.The problems considered are of two general flavors. One is a kind of a structure theory of

    set addition that is primarily concerned with identifying sets characterized by some extremalproperties, e.g. a small doubling. The doubling is defined, for any finite subset A of anabelian group, to be |A + A|/|A|. In this respect we investigate the Green-Ruzsa theoremwhich almost completely characterizes sets with this property. In particular, we prove thefirst linear bound on the dimension of the resulting progression.

    The other subject of our interest is analysis of linear equations: finding quantitativeconditions on solvability of non-invariant equations and counting the solutions thereof. Inthis regard we prove the first tight upper bounds on Ramsey-type numbers for general linearequations and prove Schinzels conjecture on the number of solutions to a linear equation incyclic groups.

    Streszczenie

    Praca prezentuje kilka wynikw dotyczcych addytywnych waciwoci skoczonych zbiorww grupach przemiennych. Obiektem naszego szczeglnego zainteresowania bd zwaszczazbiory sum (ang. sumsets) okrelone dla podzbiorw A,B dowolnej grupy przemiennej jakoA+B = {a+ b : a A, b B}.

    Rozwaane zagadnienia s dwojakiego rodzaju. Jedne stanowi rodzaj strukturalnejteorii arytmetyki zbiorw i za cel stawiaj sobie moliwie dokadn charakteryzacj zbiorwokrelonych poprzez pewne ekstremalne wasnoci. W naszym wypadku bd to zbioryo niewielkim wspczynniku podwojenia (ang. doubling), ktry jest zdefiniowany dla dowol-nego skoczonego podzbioru A grupy przemiennej jakoK(A) = |A+A|/|A|. W zwizku z tymzagadnieniem badamy twierdzenie Greena-Ruzsy, ktre niemal cakowicie charakteryzujezbiory o niewielkim wspczynniku podwojenia. W szczeglnoci, dowodzimy pierwszegoliniowego ograniczenia na wymiar cigu w tym twierdzeniu.

    Drugim obszarem naszego zainteresowania jest analiza rwna liniowych w grupach prze-miennych, a celem okrelenie warunkw istnienia (nietrywialnych) rozwiza tych rwnalub oszacowanie liczby tych rozwiza. W pracy dowodzimy pierwszego wolno rosncegogrnego ograniczenia na wielko liczb typu Ramseya zwizanych z oglnymi rwnaniamiliniowymi. Przedstawiamy rwnie dowd hipotezy Schinzla, zwizanej z liczb rozwizarwna liniowych w grupach cyklicznych.

  • Keywords

    additive combinatorics, Freiman theorem, Green-Ruzsa theorem, linear equations, Rado the-orem, arithmetic Ramsey problems, Schur numbers, Schinzel conjecture

    AMS Classification

    11B30 Arithmetic combinatorics; higher degree uniformity11D79 Congruences in many variables11P70 Inverse problems of additive number theory, including sumsets

    2

  • Contents

    Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1. Additive problems and additive combinatorics . . . . . . . . . . . . . . . . . . 7

    1.1.1. Schurs approach to Fermats Last Theorem . . . . . . . . . . . . . . . 81.1.2. Schnirelmanns approach to the Goldbach conjecture . . . . . . . . . . 9

    1.2. The problems of our interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1. Sets with small doubling . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2. Non-invariant linear equations . . . . . . . . . . . . . . . . . . . . . . 10

    1.3. Additive combinatorics beyond our interest . . . . . . . . . . . . . . . . . . . 111.3.1. Sum-product estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2. Relative results and higher order structures . . . . . . . . . . . . . . . 121.3.3. Yet broader perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2. Some basic concepts of theory of set addition . . . . . . . . . . . . . . . . . 15

    3. Freimans and Green-Ruzsas theorems . . . . . . . . . . . . . . . . . . . . . 193.1. Ruzsas approach to Freimans-type theorems . . . . . . . . . . . . . . . . . . 193.2. Green-Ruzsas theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3. Geometry of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4. Projections, the main argument . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5. Further refinement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    4. Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    5. Rado numbers and solving linear equations . . . . . . . . . . . . . . . . . . . 355.1. Classification of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . 355.2. Rado numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    5.2.1. Sketch of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2.2. Main results based on Bohr sets analysis . . . . . . . . . . . . . . . . . 41

    5.3. Schur-like numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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  • 6. Schinzels problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.1. Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2. Notation and a sketch of the argument . . . . . . . . . . . . . . . . . . . . . . 576.3. Boundary cases lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4. Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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  • Notation

    C Absolute constants, that may differ between occurrences, will be occasionallydenoted by C.

    x = minyZ |x y|lcm, gcd least common multiple, greatest common divisor

    [n] [n] = {1, 2, . . . , n} for every n NP P = {2, 3, 5, . . .}, the set of primes

    G an abelian groupZn Zn = Z/nZ

    P1, P2, . . . arithmetic progressions, i.e. sets of the form {x0 + id}Li=0 in GSpan(X) Span(X) =

    { xX

    xx : x {1, 0, 1}}

    AB, sumset AB = {a b : a A, b B} for any A,B GkA lA, iterated sumset We extend the above definition in a natural way, i.e. kA = A+ +A

    k

    aA aA = {ax : x A} for a Z and A G

    We emphasize the difference, that we shall constantly preserve, between thetwo above notions.

    K(A), doubling K(A) = |A+A|/|A|d(P ) for a generalized arithmetic progression P = P1 + +Pd we write d(P ) = d.

    Note that the above definition depends on particular representation of P .

    A,A() We identify a set A G with its indicator function A(x) = 1 if x A,A(x) = 0 otherwise.

    f , Fourier coefficient f(r) =xZ/NZ f(x)e2ixr/N for any f : Z/NZ C and r Z/NZ

    The inversion formula states that f(x) = 1NrZ/NZ f(r)e2ixr/N .

    f g, convolution (f g)(x) =tZ/NZ f(t)g(x t) for any f, g : Z/NZ C and x Z/NZ

    The convolution theorem states that (A B)(r) = A(r)B(r).

    5

  • In particular, the number of solutions to a1x1 + + akxk = 0 in A Z/NZ

    is (a1A . . . akA)(0) =1N

    rZ/NZ

    A(a1r) . . . A(akr)

    Spec(A), large spectrum Spec(A) = {r Z/NZ : |A(r)| > |A|}

    B For Bohr set B = B(, ) we write B = B(, )

    o(), O(), For positive functions f, g we define the asymptotic notations f = o(g),f = O(g) and f g to mean lim fg = 0, lim sup

    fg < and f = O(g),

    respectively.Note that the precise meaning of these symbols depends on the particular limitchosen. In our considerations it is usually in the infinity for natural-valuedparameters like n and in 0+ for real-valued , > 0. In every case the meaningis clear from context.

    6

  • Chapter 1

    Introduction

    In this thesis we shall present some results concerning additive properties of finite sets inabelian groups. It will be of primary importance to us to consider the sumsets

    A+B = {a+ b : a A, b B}

    for subsets A,B of an abelian group.The problems considered are of two general flavors. One is a kind of a structure theory of

    set addition that is primarily concerned with identifying sets characterized by some extremalproperties, e.g. a small doubling. The doubling is defined, for any finite subset A of anabelian group, to be |A + A|/|A|. In this respect we investigate the Green-Ruzsa theoremwhich almost completely characterizes sets with this property. In particular, we prove thefirst linear bound on the dimension of the resulting progression.

    The other subject of our interest is analysis of linear equations: finding quantitativeconditions on solvability of non-invariant equations and counting the solutions thereof. Inthis regard we prove the first tight upper bounds on Ramsey-type numbers for general linearequations and prove Schinzels conjecture on the number of solutions to a linear equation incyclic groups.

    A huge part of the thesis touches upon a recently developed, and still rapidly developing,field of additive combinatori