Riesz sequences and arithmetic progressions

Itay Londner¹ ; Alexander Olevskiĭ¹

¹ School of Mathematical Sciences Tel-Aviv University Tel-Aviv 69978, Israel

Studia Mathematica, Tome 225 (2014) no. 2, pp. 183-191

Résumé

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.

Détail

DOI : 10.4064/sm225-2-5

Keywords: given set mathcal positive measure circle set varlambda integers ask whether varlambda lambda lambda varlambda riesz sequence mathcal consider question connection arithmetic properties set nbsp varlambda improving result bownik speegle construct set mathcal varlambda never riesz sequence varlambda contains arithmetic progression length nbsp step ell varepsilon arbitrarily large other prove every set nbsp mathcal admits riesz sequence varlambda varlambda does contain arithmetic progressions length nbsp step ell arbitrarily large