On arithmetic sums of Ahlfors-regular sets

TL;DR

This paper establishes a lower bound on the Hausdorff dimension of the sum of two Ahlfors-regular sets in the real line, valid for all but a measure-zero set of parameters, advancing understanding of sumset dimensions.

Contribution

It provides a new lower bound on the Hausdorff dimension of sums of Ahlfors-regular sets, valid for almost all scalar multipliers, extending previous sumset dimension results.

Findings

Lower bound on $ ext{dim}_H(A + heta B)$ for almost all $ heta$

Extension of sumset dimension theory to Ahlfors-regular sets

Almost everywhere validity outside a measure-zero set

Abstract

Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: \alpha$ and $\dim_{\mathrm{H}} B =: \beta$. I prove that $$\dim_{\mathrm{H}} [A + \theta B] \geq \alpha + \beta \cdot \tfrac{1 - \alpha}{2 - \alpha}$$ for all $\theta \in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$.