A note on the sum-product problem for fractal sets

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.