A Note on Sidon Sets in Bounded Orthonormal Systems
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2018, Springer Science+Business Media, LLC, part of Springer Nature. We give a simple example of an n-tuple of orthonormal elements in L2 (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant n. This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant logn. We also include the analogous n n-matrix valued example, for which the optimal constant is 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 n. This is again asymptotically optimal. We show that any martingale difference sequence with values in [- 1 , 1] is dominated in a natural sense (related to our results) by any sequence of independent, identically distributed, symmetric { - 1 , 1 } -valued variables (e.g. the Rademacher functions). We include a self-contained proof that any sequence (n) that is the union of two Sidon sequences lying in orthogonal subspaces is such that (n n n n) is Sidon.