On the distribution of Sidon series

Abstract

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure $\mu$ and dual group $\Gamma$. Let E be a Sidon subset of $\Gamma$ with Sidon constant S(E). Let r_n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all $\alpha > 0$: c^{-1}P[| \sum_{n=1}^Na_nr_n| >= c \alpha ] <= \mu[| \sum_{n=1}^Na_n\gamma_n| >= \alpha ] <= cP [|\sum_{n=1}^Na_nr_n| >= c^{-1} \alpha ]