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Riesz sequences and arithmetic progressions

Tom 225 / 2014

Itay Londner, Alexander Olevskiĭ Studia Mathematica 225 (2014), 183-191 MSC: Primary 42C15; Secondary 42A38. DOI: 10.4064/sm225-2-5

Streszczenie

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.

Autorzy

  • Itay LondnerSchool of Mathematical Sciences
    Tel-Aviv University
    Tel-Aviv 69978, Israel
    e-mail
  • Alexander OlevskiĭSchool of Mathematical Sciences
    Tel-Aviv University
    Tel-Aviv 69978, Israel
    e-mail

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