This is an announcement for the paper “A note on Sidon sets in bounded orthonormal systems” by Gilles Pisierhttps://arxiv.org/find/math/1/au:+Pisier_G/0/1/0/all/0/1.

Abstract: We give a simple example of an $n$-tuple of orthonormal elements in $L_2$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $\approx\sqrt{n}$. This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant $\approx\sqrt{\log n}$. We deduce from our example that there are two $n$-tuples each Sidon with constant $1$, lying in orthogonal linear subspaces and such that their union is Sidon only with constant $\approx\sqrt{n}$. This is again asymptotically optimal.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.02969