A note on the sum-product problem for fractal sets

TL;DR

This paper explores the sum-product problem for fractal sets, establishing lower bounds on the dimensions of sum and product sets using recent geometric and sum-product advances.

Contribution

It introduces new bounds on the dimensions of sum and product sets for fractal sets, improving previous results by applying recent incidence geometry and sum-product theory.

Findings

Either the upper-box dimension of the product set or the lower-box dimension of the sum set exceeds a specific bound.

Improved bounds are achieved when replacing the sum-set with the difference-set.

The results apply to sets with Hausdorff dimension up to 1/2.

Abstract

Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the upper-box dimension of $AA$, or the lower-box dimension of $A+A$ must be at least $29s/23$. We obtain the slightly better bound of $33 s / 26$ when we replace the sum-set with the smoother difference-set.