A Note on the Sum-Product Problem and the Convex Sumset Problem

TL;DR

This paper improves known exponents related to the sum-product and convex sumset problems over the real numbers, providing sharper bounds for sumsets and product sets of finite and convex sets.

Contribution

It introduces new exponents for the sum-product conjecture and sumset bounds for convex sets, advancing the understanding of additive and multiplicative structures in real sets.

Findings

New exponent for sum-product conjecture: rac{4}{3} + rac{10}{4407} - \u03b5

Enhanced bounds for convex sets: rac{46}{29} - \u03b5 for sumsets, rac{8}{5} + rac{1}{3440} - rac{5} for difference sets

Abstract

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - \epsilon} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[ \left\lvert A+A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{46}{29} - \epsilon}, \] and \[ \left\lvert A-A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -\epsilon} .\]