Ilya D. Shkredov

On Sum Sets of Sets Having Small Product Set

We improve the sum–product result of Solymosi in R; namely, we prove that max{|A + A|, |AA|} ****** |A|4/3+c, where c > 0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A ⊂ R with |AA| ≤ |A|4/3.

Variations on the sum-product problem

This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set

Some new results on higher energies

A \subset \mathbb R

ADDITIVE DIMENSION AND A THEOREM OF SANDERS

:

Some remarks on the Balog–Wooley decomposition theorem and quantities D +, D ×

\exists a \in A

Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields

such that

Sums of multiplicative characters with additive convolutions

|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| \gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}

On subgraphs of random Cayley sum graphs

.

ON SOME MULTIPLE CHARACTER SUMS

In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the notion of dual popular difference sets.

On the size of the set AA+A: ON THE SIZE OF THE SET AA+A

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.