Sum-product phenomena for Ahlfors-regular sets

TL;DR

This paper extends sum-product phenomena to Ahlfors-regular sets, leveraging recent advances to establish fractal analogues of classical sum-product bounds in real analysis.

Contribution

It introduces a sum-product result for Ahlfors-regular sets, providing a fractal analogue of Solymosi's 4/3 bound for finite real subsets.

Findings

Established a sum-product estimate for Ahlfors-regular sets

Derived a fractal analogue of Solymosi's 4/3 bound

Connected recent work of Orponen to sum-product phenomena

Abstract

We utilise the recent work of Orponen to yield a sum-product result for Ahlfors-regular sets. As a corollary, we obtain the fractal analogue of Solymosi's $4/3$-bound for finite subsets of $\mathbb{R}.$