Questions about the structure of sums of Cantor sets, as well as related questions on properties of convolutions of singular measures, appear in dynamical systems (due to persistent homoclinic tangencies and Newhouse phenomena), probabilities, number theory, and spectral theory. We will describe the recent results (joint with D.Damanik and B.Solomyak) that claim that under some natural technical conditions convolutions of measures of maximal entropy supported on dynamically dened Cantor sets in most cases (for almost all parameters in a one parameter family) are absolutely continuous. This provides a rigorous proof of absolute continuity of the density of states measure for the Square Fibonacci Hamiltonian in the low coupling regime, which was conjectured by physicists more than twenty years ago.