Overlapping self-affine sets Pablo Shmerkin 28A80self-affineself-similarBernoulli convolutions We study families of possibly overlapping self-affine sets. Our main example is a family that can be considered the self-affine version of Bernoulli convolutions and was studied, in the non-overlapping case, by F. Przytycki and M. Urba\'nski [Feliks Przytycki, Mariusz Urba\'nski, \textit{On the Hausdorff dimension of some fractal sets}, Studia Math. \textbf{93} (1989), 155--186]. We extend their results to the overlapping region and also consider some extensions and generalizations. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2718 10.1512/iumj.2006.55.2718 en Indiana Univ. Math. J. 55 (2006) 1291 - 1332 state-of-the-art mathematics http://iumj.org/access/