On common energies and sumsets

We prove an elementary additive combinatorics inequality, which says that if $A$ is a subset of an Abelian group, which has, in some strong sense, large doubling, then the difference set A-A has a large subset, which has small doubling.

We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times.

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.

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset $A\subseteq\mathbb F_p$ with $p$ prime and $|A|<0.0045p$, (i) if $|A+A|<2.59|A|-3$ and $|A|>100$, then $A$ is contained in an arithmetic progression of size $|A+A|-|A|+1$, and (ii) if $|A-A|<2.6|A|-3$, then $A$ is contained in an arithmetic progression of size $|A-A|-|A|+1$. The improvement comes from using the properties of higher energies.

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.

This is a survey of old and new problems and results in additive number theory.

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 $\lambda > 2$ and every $k \geqslant (\log n)^4$: if $\omega \to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant \lambda k$ are contained in an arithmetic progression of length $...

We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is $k=k(\varepsilon)$ such that the higher energy ${\mathsf{E}}_k(A)$ is at most $|A|^{k+\varepsilon}$ unless $A$ has a very specific structure. As an application we obtain that any finite subset $A$ of the real numbers or the prime field either contains an additive Sidon--type subset of size $|A|^{1/2+c}$ or a multiplicative Sidon--type subset of size $|A|^{1/2+c}$.

In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and this is just one example of a large family of results. Our aim in this paper is to investigate what happens if we restrict the number of elements of one set that we may use to form the sums. Here is the question we set out to answer: given ...

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $\lambda X + \eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinato...