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 ⊂ R A,B,C \subset \mathbb {R} be Borel sets with 0 > dim H B ≤ dim H A > 1 0 > {\dim _{\mathrm {H}}} B \leq {\dim _{\mathrm {H}}} A > 1 and dim H B + dim H C > dim H A {\dim _{\mathrm {H}}} B + {\dim _{\mathrm {H}}} C > {\dim _{\mathrm {H}}} A . Then, there exists c ∈ C c \in C such that dim H ( A + c B ) > dim H A . \begin{equation*} {\dim _{\mathrm {H}}} (A + cB) > {\dim _{\mathrm {H}}} A. \end{equation*} We use this to show that every ( s ,