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