On the discretised $ABC$ sum-product problem

Abstract

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 ${\dim }_{\mathrm{H}}B=\beta$: $${\dim }_{\mathrm{H}}\{c\in \mathbb{R}:{\dim }_{\mathrm{H}}(A+cB)\le \alpha +\eta \}\le \frac{\alpha -\beta }{1-\beta }+\kappa .$$ 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 $a_{1}B+\dots +a_{n}B$.

55 pages pages. v3: Referee comments incorporated, to appear in Trans. Amer. Math. Soc. This version of the paper also contains the results from arXiv:2201.00564