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

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 $$\dim_{\mathrm{H}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$