LAUSR

We give a simple example of an n-tuple of orthonormal elements in $$L_2$$L2 (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $$\approx \sqrt{n}$$≈n. This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant $$\approx \sqrt{\log n}$$≈logn. We also include the analogous $$n\times n$$n×n-matrix valued example, for which the optimal constant is $$\approx n$$≈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 $$\approx \sqrt{n}$$≈n. This is again asymptotically optimal. We show that any martingale difference sequence with values in $$[-1,1]$$[-1,1] is “dominated” in a natural sense (related to our results) by any sequence of independent, identically distributed, symmetric $$\{-1,1\}$${-1,1}-valued variables (e.g. the Rademacher functions). We include a self-contained proof that any sequence $$(\varphi _n)$$(φn) that is the union of two Sidon sequences lying in orthogonal subspaces is such that $$(\varphi _n\otimes \varphi _n \otimes \varphi _n\otimes \varphi _n)$$(φn⊗φn⊗φn⊗φn) is Sidon.