Variations on the sum-product problem II

Abstract

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, and build on the methods therein. The main new results is that, for any finite set $A\subset \mathbb{R}$, there exists $a\in A$ such that $|A(A+a)|\gtrsim |A{|}^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4{)}^2+\log a_5:a_i\in A\}|\gg \frac{|A{|}^2}{\log |A|}$. The latter result is optimal up to the logarithmic factor.

arXiv commentThis paper supersedes arXiv:1603.06827