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

Abstract: We make progress on several interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, exceptional estimates for orthogonal projections, and the dimension of Furstenberg sets. We give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 &lt; {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A &lt; 1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C &gt; {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $${\dim_{\mathrm{H}}} (A + cB) &gt; {\dim_{\mathrm{H}}} A. $$ Here we only mention special cases of our results on projections and Furstenberg sets. We prove that every $s$-Furstenberg set $F \subset \mathbb{R}<sup>{2}$ has Hausdorff dimension $${\dim_{\mathrm{H}}} F \geq \max{ 2s + (1 - s)^{2}/(2 - s), 1+s}.$$ We prove that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}<sup>{2}$ associated with a $t$-Ahlfors-regular line set has $${\dim_{\mathrm{H}}} F \geq \min\left{s + t,\tfrac{3s + t}{2},s + 1\right}.$$ Let $\pi_{\theta}$ denote projection onto the line spanned by $\theta\in S<sup>1$. We prove that if $K \subset \mathbb{R}<sup>{2}$ is a Borel set with ${\dim_{\mathrm{H}}}(K)\le 1$, then $$ {\dim_{\mathrm{H}}} {\theta \in S^{1} : {\dim_{\mathrm{H}}} \pi_{\theta}(K) &lt; u} \leq \max{ 2(2u - {\dim_{\mathrm{H}}} K),0}, $$ whenever $u \leq {\dim_{\mathrm{H}}} K$, and the factor "$2$" on the right-hand side can be omitted if $K$ is Ahlfors-regular.