A Note on Sidon Sets in Bounded Orthonormal Systems Academic Article uri icon

abstract

  • © 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 (φn) that is the union of two Sidon sequences lying in orthogonal subspaces is such that (φn⊗ φn⊗ φn⊗ φn) is Sidon.

author list (cited authors)

  • Pisier, G.

citation count

  • 0

publication date

  • February 2018