Projections, Furstenberg sets, and the $ABC$ sum-product problem

By Tuomas Orponen, Pablo Shmerkin. 2023

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We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, and the dimension of Furstenberg sets. Along the way, we obtain new information on the dimension of exceptional sets of orthogonal projections. First, we give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0<{\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A<1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C>{\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $$ \dim_{\mathrm{H}} (A + cB)>{\dim_{\mathrm{H}}} A. $$ We use this to show that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a line set of equal Hausdorff and packing dimension $t$ satisfies $$\dim_{\mathrm{H}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$

View Paper: https://www.semanticscholar.org/paper/af835172806c18dc74a89812eefeee9ddb6428c8