Hausdorff dimension bounds for the ABC sum-product problem

TL;DR

This paper completes the proof of bounds on the Hausdorff dimension of sumsets involving Borel sets, extending Bourgain's 2010 result from the case of equal dimensions to the general case with different dimensions.

Contribution

It extends Bourgain's 2010 result to the case where the dimensions of the sets differ, providing new bounds on the Hausdorff dimension of sumsets involving these sets.

Findings

Established Hausdorff dimension bounds for sumsets with different set dimensions.

Extended Bourgain's 2010 result from equal to unequal set dimensions.

Demonstrated that box dimension results imply Hausdorff dimension results at the discretized level.

Abstract

The purpose of this paper is to complete the proof of the following result. Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. Then, there exists $\eta > 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A = \alpha$ and $\dim_{\mathrm{H}} B = \beta$, then $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq \alpha + \eta\} \leq \tfrac{\alpha - \beta}{1 - \beta} + \kappa.$$ This extends a result of Bourgain from 2010, which contained the case $\alpha = \beta$. This paper is a sequel to the author's previous work from 2021 which, roughly speaking, established the same result with $\dim_{\mathrm{H}} (A + cB)$ replaced by $\dim_{\mathrm{B}}(A + cB)$, the box dimension of $A + cB$. It turns out that, at the level of $\delta$-discretised statements, the superficially weaker box dimension result formally implies the Hausdorff dimension result.