SELF-SIMILAR MEASURES AND INTERSECTIONS OF CANTOR SETS

It is natural to expect that the arithmetic sum of two Canter sets sho uld have positive Lebesgue measure if the sum of their dimensions exce eds 1, but there are many known counterexamples, e.g, when both sets a re the middle-alpha Canter set and alpha is an element of (1/3, 1/2). We show that for any compact set K and for a.e. alpha is an element of (0, 1), the arithmetic sum of K and the middle-alpha Canter set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds I. In this case we also determine the essential sup remum, as the translation parameter t varies, of the dimension of the intersection of K + t with the middle-alpha Canter set. We also establ ish a new property of the infinite Bernoulli convolutions upsilon(lamb da)(p) (the distributions of random series Sigma(n=0)(infinity) +/-lam bda(n) where the signs are chosen independently with probabilities (p, I - p)). Let 1 less than or equal to q(1) < q(2) less than or equal t o 2. For p not equal 1/2 near 1/2 and for a.e. lambda in some nonempty interval, upsilon(lambda)(p) is absolutely continuous and its density is in L-q1 but not in L-q2. We also answer a question of Kahane conce rning the Fourier transform of upsilon(lambda)(1/2).