A new expander and improved bounds for $A(A+A)$

Abstract

The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4{)}^2+\log a_5:a_1,a_2,a_3,a_4,a_5\in A\}|\gg \frac{|A{|}^2}{\log |A|}.$$ This bound is optimal, up to logarithmic factors. The paper also gives new lower bounds for $|A(A-A)|$ and $|A(A+A)|$, improving on results from arXiv:1312.6438. The new bounds are $$|A(A-A)|⪆|A{|}^{3/2+\frac{1}{34}}$$ and $$|A(A+A)|⪆|A{|}^{3/2+\frac{5}{242}}.$$

This paper has now been superseded by arXiv:1703.09549

References in corpus (2)

• On sums of Szemerédi--Trotter sets
• New results on sum-products in R

Cited by in corpus (1)

• Expanders with superquadratic growth