Hausdorff dimension bounds for the ABC sum-product problem

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.