Additive dimension and a theorem of Sanders

Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadrati...

We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.

Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.

We obtain a non--trivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to growth of conjugates classes, estimates of exponential sums and the restriction phenomenon.

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of prope...

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 torsion-free or elementary abelian then |C|\geq |A|+|B|-|S| -m. We also prove that |C|\geq |A|+|B|-2|S|-m if the torsion subgroup of G is cyclic. In the case S={0} this provides an advance on a conjecture of Lev.

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.

Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of $A$ is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime $|A| = |G|^\alpha $ whenever $\alpha \le c$, where $c \ge 1/2$ is some absolute constant. On the other hand, there are statistical results, which apply only to a noticeable fraction of the elements, which give nontrivial bounds even to much sm...

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. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = Z_{2n}, and who obtained a weaker threshold (up to a constant factor) in general.

For a positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.