On the discretised $ABC$ sum-product problem

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 $a_{1}B + \ldots + a_{n}B$.