Self-similar measures and intersections of Cantor sets

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-ca Cantor set and Ca E (1, 1). We show that for any compact set K and for a.e. ag E (0, 1), the arithmetic sum of K and the middle-ae Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential su-premum, as the translation parameter t varies, of the dimension of the intersection of K + t with the middle-ca Cantor set. We also establish a new property of the infinite Bernoutlli convolutions lJ (the distributions of random series E' 0 ?A', where the signs are chosen independently with probabilities (p,1 p)). Let 1 < ql < q2 < 2. For p 2 near 2 and for a.e. A in some nonempty interval, IV is absolutely continuous 2 1 and its density is in LIl but not in Lq2. We also answer a question of Kahane concerning the Fourier transform of vl/2

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