A superadditivity and submultiplicativity property for cardinalities of sumsets

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.

In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+\epsilon}}.

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i...

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such that if $m\ge\max(A)-|A|+2$, then the $m$-fold sumset $mA$ consists of the union of these sets and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and investigate the corresponding st...

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\ldots,k-1$. In this paper, we give the best lower bound for $|h^{(\mathbf{r})}A|$ in terms of $\mathbf{r}...

In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the additive energy of multiplicative subgroups and convex sets and also a series another results on the connection of the additive energy and so--called higher moments of convolutions. Besides we prove new theorems on multiplicative subgroups concerning lower bounds for its doubling consta...

For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA =\{a_1+... a_l| a_i \in A_i \}. $$ A closely related and equally interesting notion is that of $l^{\ast}A$, which is the collection of numbers which can be represented as a sum of $l$ different elements of $A$ $$l^{\ast} A =\{a_1+... a_l| a_i \in A_i, a_i \neq a_j \}. $$ The goal of this paper is to investigate the structure of $l...

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s dist...