Variations on the Sum-Product Problem

TL;DR

This paper advances sum-product problem bounds in real sets, providing new lower bounds for expressions like |A(A+A)| and |A(A+A+A+A)|, and demonstrating typical expanders with significant growth.

Contribution

It introduces improved bounds for sum-product expressions and expands understanding of sum-product estimates and expanders in real sets.

Findings

|A(A+A)|  |A|^{3/2 + 1/178}

|A(A+A+A+A)|  |A|^2 / \u221a{}(|A|)

|A(A+a)|  |A|^{3/2} for typical a

Abstract

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a 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 that $$|A(A+a)|\gg{|A|^{3/2}}$$ holds for a typical element of $A$.