Since addition is commutative but subtraction is not, the sumset S+S of a finite set S is predisposed to be smaller than the difference set S-S. In this paper, however, we show that each of the three possibilities (|S+S|>|S-S|, |S+S|=|S-S|, |S+S|<|S-S|) occur for a positive proportion of the subsets of {0, 1, ..., n-1}. We also show that the difference |S+S| - |S-S| can take any integer value, and we show that the expected number of omitted differences is asymptotically 6 while the expected number of missing sums is asymptotically 10. Other data and conjectures on the distribution of these quantities are also given.
We prove new general results on sumsets of sets having Szemerédi--Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.
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 $A$.
We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.
Let $A$ be a subset of $\mathbb{Z} / N\mathbb{Z}$ and let $\mathcal{R}$ be the set of large Fourier coefficients of $A$. Properties of $\mathcal{R}$ have been studied in works of M.-- C. Chang, B. Green and the author. In the paper we obtain some new results on sets of large exponential sums.
In the paper we are studying some properties of subsets Q of sums of dissociated sets. The exact upper bound for the number of solutions of the following equation (1) q_1 + ... + q_p = q_{p+1} + ... + q_{2p}, q_i \in Q in groups F_2^n is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of …
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.
We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.
In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [24] and several of subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.
Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small product with $A$.
Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has "more sums than differences" if |A+A|>|A-A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.
Suppose that P is an infinite set of primes such that P = A + B + C, where A,B,C are sets with at least two elements. We show that if P(x) > c x/log^d x (where P(x) = the number of elements of P that are <= x), and if A,B,C is a "regular" triple of sets, then either |A+B| <= d, or |B+C| <= d, or |A+C| <= d.
If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.
Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.
We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that, except for a few cases with small $d$, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ra…
In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $λ> 2$ and every $k \geqslant (\log n)^4$: if $ω\to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant λk$ are contained in an arithmetic progression of length $λk/2 + ω$.
We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
In this paper we show that if $A$ is a subset of the primes with positive relative density $δ$, then $A+A$ must have positive upper density $C_1δe^{-C_2(\log(1/δ))^{2/3}(\log\log(1/δ))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.
Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 + \frac{1}{d-1} \bigg) |A| - O_{d}(|A|^{1- δ}), \] for some absolute constant $δ>0$ that only depends on $d$. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an $O_{d}(|A|^{1- δ})$ error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.
We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we obtain new results on incomplete multilinear Kloosterman sums in finite fields.
Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<α|A|$ and $α\ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\times$ is …
Let A be a subset of Z / NZ, and let R be the set of large Fourier coefficients of A. Properties of R have been studied in works of M.-C. Chang and B. Green. Our result is the following : the number of quadruples (r_1, r_2, r_3, r_4) \in R^4 such that r_1 + r_2 = r_3 + r_4 is at least |R|^{2+ε}, ε>0. This statement shows that the set R is highly structured. We also discuss some of the generalizations and applications of our result.
We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.
We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of …
A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This improves the previous result of Martin and O'Bryant that there is a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx 4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to arbitrary precision. We also describe the structure of a random MSTD subset S of {0, 1, ..., n}. We formalize the intuition that …
In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.
\medskip\noindent\textbf{Résumé.} Soit $l$ un entier et $\ors=(s_1, \dots, s_l)$ une séquence d'entiers positifs. Dans ce document, nous étudierons les propriétés arithmétique de sommes harmoniques multiples $H(\ors; n)$, qui est le $n$-ème somme partielle de la valeur de la série multiple zeta $ζ(\ors)$. On conjecture que pour tout $\ors$ et de tous les premiers $p$, il n'y a que de nombreux finitely $p$-partie intégrante sommes $H(\ors,n)$. Ceci généralise une conjecture de Eswarathasan et Levine et Boyd pour la série harmonique. Nous fournissons beaucoup d'éléments de preuve pour cette conj…
We obtain quantitative estimates for the asymptotic density of subsets of the two-dimensional integer lattice which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov's mean value theorem applicable to binary forms.
For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $Σ_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small, $|Σ_n(S)|\leq |S|-n+1$, it is known that the terms of $S$ can be partitioned into $n$ nonempty sets $A_1,\ldots,A_n\subseteq G$ such that $Σ_n(S)=A_1+\ldots+A_n$. Moreover, if the upper bound is strict, then $|A_i\setminus Z|\leq 1$ for all $i$, where $Z=\bigcap_{i=1}^{n}(A_i+H)$ and $H=\{g\in G:\; g+Σ_n(S)=Σ_n(S)\}$ is the stabilizer of $Σ_n(S)$. This allows structu…
We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Polya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.
We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum functi…
Let $h$ be a positive integer and let $\varepsilon > 0$. The Haight-Ruzsa method produces a positive integer $m^*$ and a subset $A$ of the additive abelian group $\mathbf{Z}/m^*\mathbf{Z}$ such that the difference set is large in the sense that $A-A = \mathbf{Z}/m^*\mathbf{Z}$ and $h$-fold sumset is small in the sense that $|hA| < \varepsilon m^*$. This note describes, and in a modest way extends, the Haight-Ruzsa argument, and constructs sets with more differences than multiple sums in other additive abelian groups.
We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…
Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of (Z_2)^n. These cosets are arranged as Hamming balls, the smaller of which has radius 1. By similar methods, we re-prove the Freiman-Ruzsa theorem in (Z_2)^n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |<A>|/|A|, over all subsets A of (Z_2)^n with doubling constant |A+A|/|A| < K. …
We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square--root barrier in a problem of Balog concerning products of differences in a field of prime order.
Every set $A$ of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that $B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I} B_i \subseteq A$ for every nonempty finite set $I$ of positive integers.
The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plunnecke's inequality.
Given a set A in Z/NZ we may form a Cayley sum graph G_A on vertex set Z/NZ by joining i to j if and only if i + j is in A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G_A is a.s. O(log N). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on Z/NZ. Indeed, we also show that the clique number of a random Cayley sum graph on (Z/2Z)^n, 2^n = N, is almost surely not O(log N). Despi…
We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and Sárközy.
