Nakhlé Asmar and Stephen Montgomery-Smith, On the distribution of Sidon series. Arkiv Mat. 31, (1993), 13-26. 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 \(\alpha>0\): \[ c^{-1} P\left[\left|\sum a_n r_n\right| \ge c \alpha\right] \le \mu\left[\left|\sum a_n \gamma_n \right| \ge \alpha \right] \le c P\left[\left|\sum a_n r_n\right| \ge c^{-1} \alpha\right] \] where \(a_1,\dots,a_N\) are arbitrary elements of \(B\), and \(\gamma_1,\dots,\gamma_N\) are arbitrary elements of \(E\). We prove a similar result for Sidon subsets of dual objects of compact groups, and apply our results to obtain new lower bounds for the distribution functions of scalar-valued Sidon series. We also note that either one of the above inequalities, even in the scalar case, characterizes Sidon sets. (tex, dvi, ps, pdf.)


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