A Note on Sidon Sets in Bounded Orthonormal Systems

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 ≈ n . This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant ≈ log n . 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 ( φ n ) that is the union of two Sidon sequences lying in orthogonal subspaces is such that ( φ n ⊗ φ n ⊗ φ n ⊗ φ n ) is Sidon.