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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Riesz sequences and arithmetic progressions

### Tom 225 / 2014

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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