Riesz sequences and arithmetic progressions
Volume 225 / 2014
Abstract
Given a set $\mathcal {S}$ of positive measure on the circle and a set $\varLambda $ of integers, one can ask whether $E(\varLambda ):=\{ e^{i\lambda t}\} _{\lambda \in \varLambda }$ is a Riesz sequence in $L^{2}(\mathcal {S})$.
We consider this question in connection with some arithmetic properties of the set $\varLambda $. Improving a result of Bownik and Speegle (2006), we construct a set $\mathcal {S}$ such that $E(\varLambda )$ is never a Riesz sequence if $\varLambda $ contains an arithmetic progression of length $N$ and step $\ell =O(N^{1-\varepsilon })$ with $N$ arbitrarily large. On the other hand, we prove that every set $\mathcal {S}$ admits a Riesz sequence $E(\varLambda )$ such that $\varLambda $ does contain arithmetic progressions of length $N$ and step $\ell =O(N)$ with $N$ arbitrarily large.
