On an application of higher energies to Sidon sets

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper, answer Szemer\'edi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\epsilon}$ for some $\epsilon>0$. In fact, this...

A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains a symmetric subset with measure d, while if d > D(x), then there exists a subset of [0,1] with measure x that does not contain a symmetric subset with measure d. In this paper we establish upper and lower bounds for D(x) of the same order ...

These notes basically contain a material of two mini--courses which were read in G\"{o}teborg in April 2015 during the author visit of Chalmers & G\"{o}teborg universities and in Beijing in November 2015 during "Chinese--Russian Workshop on Exponential Sums and Sumsets". The article is a short introduction to a new area of Additive Combinatorics which is connected which so--called the higher sumsets as well as with the higher energies. We hope the notes will be helpful for a ...

A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size $O((n \cdot 2^n)^{1/3})$ in the group $\mathbb{Z}_2^n$, generalizing a result of Ruzsa concerning maximal Sidon sets in the integers.

In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.

We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.

We prove that maximal cardinality of the Sidon set from $[n]$ exceeds $\sqrt{n}

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1,2,...,n} are B_2[g]-sets, in the sense that if B_2[g](k,n) denotes the number of B_2[g]-sets of cardinality ...

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.

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality a...