Related papers: Most Subsets are Balanced in Finite Groups

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…

The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements…

Let $G$ be a discrete abelian group. F{\o}lner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach…

Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical…

Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such…

Fermat's statement is equivalent to say that if $x$, $y$, $z$, $n$ are integers and $n>2$, then $z^{n}\gtrless x^{n}+y^{n}$. This is proved with the aid of numbers $\lambda $'s, of the form $\lambda =z/\rho $, with $1<\rho<z$, named…

In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the…

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

One form of the inclusion-exclusion principle asserts that if A and B are functions of finite sets then A(S) is the sum of B(T) over all subsets T of S if and only if B(S) is the sum of (-1)^|S-T| A(T) over all subsets T of S. If we replace…

We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of…

Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…

Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of…

A More Sums Than Differences (MSTD) set is a finite set of integers $A$ where the cardinality of its sumset, $A+A$, is greater than the cardinality of its difference set, $A-A$. We address a problem posed by Samuel Allen Alexander that asks…

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$…

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent…

The solubilizer of an element $x$ of a profinite group $G$ is the set of the elements $y$ of $G$ such that the subgroup of $G$ generated by $x$ and $y$ is prosoluble. We propose the following conjecture: the solubilizer of $x$ in $G$ has…

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group.…

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c > 0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A| <…