Imre Ruzsa

Alfréd Rényi Institute of Mathematics · Research Division of Number Theory

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$.

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa–Leindler inequality. This is then applied to show that if \(A, B \subseteq \mathbb {Z}^d\) are finite sets and U is a subset of a “quasicube”, then \(|A + B + U| \geqslant |A|{ }^{1/2} |B|{ }^{1/2} |U|\). Th...

Let d be a positive integer and U ⊂ ℤd finite. We study $$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B + U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| B \right|}^{1/2}}}},$$and other related quantities. We employ tensorization, which is not available for the doubling constant, ∣U + U∣...

С помощью построения подходящих неотрицательных экспоненциальных сумм получены верхние оценки размера произвольного множества $B_q$ в циклической группе $\mathbb Z_q$ такого, что множество разностей $B_q-B_q$ не содержит кубических вычетов по модулю $q$.

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this resu...

For a set A of points in the plane, not all collinear, we denote by tr(A) the number of triangles in a triangulation of A, that is, tr(A)=2i+b-2, where b and i are the numbers of boundary and interior points of the convex hull [A] of A respectively. We conjecture the following discrete analog of the Brunn–Minkowski inequality: for any two finite po...

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this resu...

This note is an addendum to an earlier paper of the authors. We describe improved constructions addressing a question of Erd\H{o}s and Szemer\'edi on sums and products of real numbers along the edges of a graph. We also add a few observations about related versions of the problem.

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Pr\'ekopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is...

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$\beta(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$\beta(U) = |U|,$$ whe...

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant.

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of t...

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ that displays block structure (i.e., is m...

A theorem of Følner asserts that for any set A ⊂ Z of positive density there is a Bohr neigbourhood B of 0 such that B \ (A − A) has zero density. We use this result deduce some consequences about the structure of difference sets of sets of integers having a positive upper density. 2000 Mathematics Subject Classification:11B75,05D10.

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x1,…,xnx1,…,xn consisting of equations of the form ∏i∈Ixi=b∏i∈Ixi=b, where xi,b∈{−1,1}xi,b∈{−1,1} and I⊆[n]I⊆[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+...

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for a...

We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard matrices (MUHs) in any dimension cannot contain more than one real Hadamard matrix. We also give new proofs of seve...

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive exponential sums using frequencies from $A$.

This short note gives an upper bound on the measure of sets $A\subset [0,1]$ such that $x+y=3z$ has no solutions in $A$.

We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard matrices (MUHs) in any dimension $d$ cannot contain more than one real Hadamard matrix. We also give new proofs of...

Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(ϵ) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x1/h-ϵ. We give two new proofs of this result. The first one consists of an explicit co...

In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over \(\mathbb{F}_2\) in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations....

The aim of this paper is to prove a general version of Plünnecke’s inequality. Namely, assume that for finite sets A, B1,...Bk we have information on the size of the sumsets A + Bi1 · · + Bil for all choices of indices i1,...il. Then we prove the existence of a non-empty subset X of A such that we have ‘good control ’ over the size of the sumset X...

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, there exists $g = g_h(\epsilon)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$ elements of $A$ is bounded by $g$, and such that $|A \cap [1,x]| \gg x^{1/h - \epsilon}$. We give two new proofs...

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon. Comment: 21 pages, no figures

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c √ n, where c> 0 is a constant independent of n. Let ℓ∆n be the n-dimensional regular simplex of edge length ℓ, and let ℓQn be the n-dimensional hypercube of edge length ℓ. For simplicity, we omit ℓ if ℓ = 1, e.g., Qn denotes the unit hypercube....

Let ℕ, ℕ0, ℤ and ℕ d denote, respectively, the sets of positive integers, non-negative integers, integers and d-dimensional integral lattice points. Let G denote an arbitrary abelian group and let X denote an arbitrary abelian semigroup, written additively. Let |S| denote the cardinality of the set S. For any sets A and B, we write A∼B if their sym...

First of all we are going to introduce what we call the polynomial lemma, a simple but very powerful result.

Our notation and terminology is consistent with [71]. We briefly gather some key notions. We denote by ℕ the set of positive integres, and we put ℕ0 = ℕ ∪ { 0 }. For real numbers, a, b ∈ ℝ we set [a, b] = { x ∈ ℤ | a ≤ x ≤ b , and we define sup Ø = max Ø = min Ø = 0.

This is a somewhat extended version of my course given in the Doc Course in Barcelona, Spring 2008. Two students, Itziar Bardají and Lluís Vena helped me to prepare the final version, and here I wish to express my sincere thanks to them.

The investigation of inverse problems has a long tradition in combinatorial number theory (see [107, 37]), and more recently it has been promoted by applications in the theory of non-unique factorizations. In this chapter we discuss the inverse problems associated with the invariants D(G), η(G) and s(G). More precisely, we investigate the structure...

Sets of lengths in Krull monoids and in noetherian domains are finite and nonempty. Furthermore, either all sets of lengths are singletons or sets of lengths may become arbitrarily large (see Lemma 1.0.1).

This is the extended and revised version of notes written for the Advanced Course in Combinatorics and Geometry: Additive Combinatorics. The course took place at the Centre de Recerca Matemàtica (CRM) at Barcelona in spring 2008. It gives a survey on the interaction between two, at first glance very disparate areas of mathematics: Non-Unique Factor...

We start with the classical theorems of Kneser and Kemperman-Scherk which are fundamental in additive group theory. Having these results at our disposal we continue the investigation of the Davenport constant, of the Erdős-Ginzburg-Ziv constant and of related invariants in zero-sum theory.

We denote by H x the set of invertible elements of H, and we say that H is reduced if H x=1. Let H red=H/H x=aH x|a ∈ H be the associated reduced monoid and q(H) a quotient group of H.

This chapter is about questions of the following kind. Assume we have finite sets A, B in a group G. What can we say about A+B if we know the structure of G, or we have some information about how these sets are situated within G? The “what” will be in most cases a lower estimate for the cardinality.

We want to describe sets that have few sums. If |A|=m, then clearly |A+A|≥m in every group (with equality for cosets), which can be improved to 2m-1 for sets of integers (or torsion-free groups in general). What can we say if we know that |A+A|≤αm, where α is constant or grows slowly as n→∞? That is, we are looking for statements of the form |A|=m,...

In this chapter we study the Davenport constant, a classical combinatorial invariant which has been investigated since the 1960s (see [115, 105, 31, 108, 103]). From the very beginning the investigation of this invariant was related also to arithmetical problems (it is reported in [108] that in 1966 H. Davenport asked for D(G), since it is the larg...

A finite set is naturally measured by its cardinality. A set of reals is naturally measured by its Lebesgue measure (non-measurable sets do exist, just we never meet them). There is no similarly universal way to measure and compare infinite sets of integers. The most naturally defined one is the asymptotic density.

The next collection of exercises is deliberately vague. The number of points is my estimate for the difficulty. This naturally depends on expertise; some of them were told in the course; then naturally the difficulty turns to 0.

Let A and B be sets in a (mostly commutative) group. We will call the group operation addition and use additive notation. The sumset of these sets is A+B=a+b:a∈ A, b∈B

Let A, B be sets in a group, |A|=m, |B|=n. The cardinality of A+B can be anywhere between max(m, n) and mn. Our aim is to understand the connection between this size and the structure of these sets.

In this chapter we mention some loosely connected things. The common feature is that we now leave the safe familiar world of finite sets. Our excursions are in two different directions.

We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times.

The aim of this paper is to prove a general version of Pl\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices $i_1, ... i_l.$ Then we prove the existence of a non-empty subset $X$ of $A$ such that we have `good control' over...

We study those functions that can be written as a finite sum of periodic integer valued functions. On Z we give three different characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a Z → Z function implies the existence of an integer valued periodic decomposition with the same periods...

For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addit...

In a given abelian group, let $A$ and $B$ be two finite subsets satisfying the small sumset condition $|A+B|\le K|A|$. We consider the problem of estimating how large $|A-B|$ can be in terms of $|A|$ and $K$ and the one of estimating the ratio $|X-B|/|X|$ when $X$ runs over all the non-empty subsets of $A$.

A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.

We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we...

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets o...

We give a non-trivial upper bound for Fh(g,N), the size of a Bh[g] subset of {1,…,N}, when g>1. In particular, we prove F2(g,N)⩽1.864(gN)1/2+1, and Fh(g,N)⩽1(1+cosh(π/h))1/h(hh!gN)1/h, h>2. On the other hand, we exhibit B2[g] subsets of {1,…,N} with g+[g/2]g+2[g/2]N1/2+o(N1/2), elements.

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p(t) such that |hA| = p(...

It is shown that every set ofnintegers contains a subset of sizeΩ(n1/6) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for everyε>0 and everym>m(ε) the following holds. IfA1, …, Amaremsubsets of cardinality at leastm1+εeach, then there area1∈A1, …, am∈Amso that the sum of every non...

We present a simple and general algebraic technique for obtaining results in Additive Number Theory, and apply it to derive various new extensions of the Cauchy-Davenport Theorem. In particular we obtain, for subsets A 0 ; A 1 ; : : : ; A k of the finite field Z p , a tight lower bound on the minimumpossible cardinality of fa 0 + a 1 + : : : + a k...

1. A Goldbach-sejtésről: Ismert a páros Golbach-sejtéssel kapcsolatban, hogy majdnem minden páros szám esetén igaz a páros Goldbach-sejtés. A pályázat keretei között a korábbi eredményeket Pintz János lényegesen megjavította, megmutatva, hogy a kivételek száma X-ig legfeljebb O(X^{2/3}). A páros Goldbach-sejtés irányában Linnik bizonyította, hogy v...

A kutatók számos érdekes eredményt értek el a kombinatorikus számelmélet és geometria, gráfelmélet, diofantikus approximáció területén, itt csak néhányat említünk. Elekes és Ruzsa a Freiman, Balog-Szemerédi és Laczkovich-Ruzsa tételek közös általánosítását adják, ezzel a témakört egységesítik, és számos kombinatorikus geometriai tételt fejlesztenek...

A kutatás keretében a 4 év alatt összesen 79 tudományos dolgozat született, melynek nagy többsége erős nemzetközi folyóiratban jelent meg. Az elért eredmények közül kiemelkednek Pintz és társszerzői prímszámelméletben elért világra szóló eredményei, melyeknek igen nagy nemzetközi visszhangja van. Jelentősek Gyarmati, Sárközy és társszerzőik pszeudo...