Variations on the Sum-Product Problem

Abstract

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set A ⊂ R,giving a partial answer to a conjecture of Balog. In a similar spirit, it is established thata bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing thatholds for a typical element of A.

On higher energy decompositions and the sum–product phenomenon

“…The work in this paper builds directly upon the works in [BaWo,El,KoRu,MRS1,RSS,ScSh,Sh1,Sh2,Sh3,So2]. It is worth noting that there are orthogonal works addressing the sum-product phenomenon.…”

“…which was addressed [MRS1] while studying the expanders from Theorem 1.14. It turns out, much like Szemerédi-Trotter is naturally a third moment estimate, their lemma is naturally a fourth moment estimate.…”

A family of four-variable expanders with quadratic growth

“…For many such functions, the images of sets are known to always grow. For example, the authors of [MRNS15] have studied several multivariable polynomials, including the function…”

On the size of the set AA+A

“…The set is just one example of a set defined by a combination of additive and multiplicative operations. Such sets have been well studied in recent years; for example, in [, ] the dual problem for the set was considered, and it was proven in that For sets formed from more variables, quantitatively better bounds, in many cases optimal up to constant and logarithmic factors, have been established. See [, ] and the references contained therein for more on such variations on the sum–product problem.…”

“…The set AA + A is just one example of a set defined by a combination of additive and multiplicative operations. Such sets have been well studied in recent years; for example, in [9,10] the dual problem for the set A(A + A) was considered, and it was proven in [10] that (1) .…”