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 $A\subset{\mathbb{R}}$, $|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$ 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 $A$.