  Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions
$\dim_{\mathrm{H}} A =: \alpha$ and $\dim_{\mathrm{H}} B =: \beta$. I prove
that $$\dim_{\mathrm{H}} [A + \theta B] \geq \alpha + \beta \cdot \tfrac{1 -
\alpha}{2 - \alpha}$$ for all $\theta \in \mathbb{R} \, \setminus \, E$, where
$\dim_{\mathrm{H}} E = 0$.
