Self-similar measures and intersections of Cantor sets

Abstract

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-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ 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 supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the infinite Bernoulli convolutions $\nu _\lambda ^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda ^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda ^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda$.