Related papers: Subset sums in abelian groups

We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.

The main result in Y.~O.~Hamidoune's paper "Adding Distinct Congruence Classes" ({\em Combin.~Probab.~Comput.}~{\bf 7} (1998) 81-87) is as follows: If $S$ is a generating subset of a cyclic group $G$ such that $0 \not \in S$ and $|S| \geq…

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log \log |G:H|}$. This…

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of…

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$…

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…

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of…

Let A,B,S be finite subsets of an abelian group G. Suppose that the restricted sumset C={a+b: a in A, b in B, and a-b not in S} is nonempty and some c in C can be written as a+b with a in A and b in B in at most m ways. We show that if G is…

In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof…

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson…

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $\sigma_1, \ldots, \sigma_H \in \mathfrak{S}_n$, there exist finite subsets $A_1,…

In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that…

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for…

Let $n$ be a large number. A subset $A$ of $Z_n$ is complete if $S_A = Z_n$, where $S_A$ is the collection of the subset sums of $A$. Olson proved that if $n$ is a prime and $|A|> 2n^{1/2} $, then $S_A$ is complete. We show that a similar…

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The $n$-term subsums version of Kneser's Theorem, obtained either via the DeVos-Goddyn-Mohar…

We provide optimal upper bounds on the growth of iterated sumsets $hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay's theorem in commutative algebra…

Let $\Gamma$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset \Gamma$ is a $C_h[g]$-set if given any set $X \subset \Gamma$ with $|X| = k$, and any set $\{ k_1 , \dots , k_g \} \subset \Gamma$, at least one of the…

For a sequence $S$ over a finite abelian group, let $MZ(S)$ denote the length of the shortest nonempty zero-sum subsequence of $S$. We prove that if $G$ is finite abelian of order $n$ and $S$ has length $n$, then $MZ(S)\le n-|\supp(S)|+1$.…

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…