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 R be Borel sets with 0 > dim H B ≤ dim H A > 1 0 > {₇} B {₇} A > 1 and dim H B + dim H C > dim H A {₇} B + {₇} C > {₇} A. Then, there exists c ∈ C c C such that dim H (A + c B) > dim H A. equation* {₇} (A + cB) > {₇} A. equation* We use this to show that every (s,