Loading…

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

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 l...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-04
Main Author: Roche-Newton, Oliver
Format: Article
Language:English
Subjects:
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 $$|A(A-A)| \gtrapprox |A|^{3/2+\frac{1}{34}}$$ and $$|A(A+A)| \gtrapprox |A|^{3/2+\frac{5}{242}}.$$
ISSN:2331-8422