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 $0<{\dim }_{\mathrm{H}}B\le {\dim }_{\mathrm{H}}A<1$ and ${\dim }_{\mathrm{H}}B+{\dim }_{\mathrm{H}}C>{\dim }_{\mathrm{H}}A$. Then, there exists $c\in C$ such that $${\dim }_{\mathrm{H}}(A+cB)>{\dim }_{\mathrm{H}}A\mathrm{.}$$ 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 $${\dim }_{\mathrm{H}}F\ge \min \{s+t,\frac{3s+t}{2},s+1\}\mathrm{.}$$