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 $A$.

30 pages, new version contains improved exponent in main theorem due to suggestion of M. Z. Garaev