On arithmetic sums of Ahlfors-regular sets

Abstract

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 $${\dim }_{\mathrm{H}}[A+\theta B]\ge \alpha +\beta \cdot \frac{1-\alpha }{2-\alpha }$$ for all $\theta \in \mathbb{R}\setminus E$, where ${\dim }_{\mathrm{H}}E=0$.

44 pages. v2: incorporated referee comments, to appear in GAFA