Antti Kaenmaki: Do self-conformal measures on the real line resonate?

In the 60's, Furstenberg conjectured that if X,Y \subset [0,1] are closed sets invariant under multiplication by integers p and q mod 1, respectively, then for any s \ne 0, the resonance inequality
  dim_H(X+sY) < min{ 1,dim_H(X) + dim_H(Y) }
implies that log(p)/log(q) is a rational number. Intuitively, if log(p)/log(q) is irrational, then X and Y dissonate by which we mean that the expansions of X and Y in bases p and q, respectively, have no common structure. A similar question can be asked for convolutions of two measures. In this talk, which is based on a recent work with B. Bárány, A. Pyörälä, and M. Wu, we answer whether two self-conformal measures in the real line resonate or dissonate.