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 $0 < β\leq α< 1$ and $κ> 0$. Then, there exists $η> 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$, then $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. 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 $δ$-discretised statements, the superficially weaker box dimension result formally implies the Hausdorff dimension result.