On the discretised ABC sum-product problem

Abstract

Let 0 < beta <= alpha < 1 and kappa > 0. I prove that there exists eta > 0 such that the following holds for every pair of Borel sets A, B subset of R with dim(H) A = alpha and dim(H) B = beta: dim(H) {c is an element of R : dim(H) (A + cB) <= alpha + eta} <= alpha-beta/1-beta + kappa. This extends a result of Bourgain from 2010, which contained the case alpha = beta. The paper also contains a delta-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form alpha B-1 + ... + alpha B-n..