Variations on the sum-product problem II

Brendan Murphy, Oliver Roche-Newton, Ilya Shkredov

Published 2017-03-28Version 1

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, as well as generalising a method therein to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: \begin{align*} \exists a \in A \text{ such that }|A(A+a)| &\gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| &\gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| &\gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| &\gg \frac{|A|^2}{\log |A|}. \end{align*}