On the discretised $ABC$ sum-product problem

Let $0<\beta \le \alpha <1$ 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 $\alpha = \beta$\mathrm{.}Thepaperalsocontainsa$\delta$-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form $a_{1}B+\dots +a_{n}B$.