A convolution of two singular continuous measures can be singular continuous or absolutely continuous (or of a mixed type). It is usually hard to determine which case is present for a specific pair of measures. It turnes out that for measures of maximal entropy of large Hausdorff dimension supported on dynamically defined Cantor sets generically the convolution is a.c. (this is a joint result with D.Damanik and B.Solomyak). This is in a sense a measure-theoretical counterpart of the claim (known as Newhouse Gap Lemma) that the sum of two sufficiently thick Cantor sets must contain an interval.