On the discretised $ABC$ sum-product problem

TL;DR

This paper extends Bourgain's 2010 result on the sum-product problem for Borel sets, providing new bounds on the Hausdorff dimension of sumsets and their discretized versions, with implications for sums of multiple scaled sets.

Contribution

It generalizes Bourgain's sum-product estimate to cases where the sets have different dimensions and introduces discretized and stronger versions of the bounds.

Findings

Bound on the Hausdorff dimension of exceptional set of translations

Discretized version of the sum-product estimate

New results on sums of multiple scaled sets

Abstract

Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. I prove that there exists $\eta > 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = \alpha$ and $\dim_{\mathrm{H}} B = \beta$: $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq \alpha + \eta\} \leq \tfrac{\alpha - \beta}{1 - \beta} + \kappa.$$ This extends a result of Bourgain from 2010, which contained the case $\alpha = \beta$. The paper also contains a $\delta$-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form $a_{1}B + \ldots + a_{n}B$.