# Variations on the Sum-Product Problem

Brendan Murphy, Oliver Roche-Newton, Ilya D. Shkredov

Research output: Contribution to journalArticle (Academic Journal)

### 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 $A$.
Original language English SIAM Journal on Discrete Mathematics Published - 22 Dec 2013

### Bibliographical note

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

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