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doi: 10.3934/dcds.2021175
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## Sublacunary sets and interpolation sets for nilsequences

 Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA

Received  August 2021 Revised  October 2021 Early access November 2021

A set $E \subset \mathbb{N}$ is an interpolation set for nilsequences if every bounded function on $E$ can be extended to a nilsequence on $\mathbb{N}$. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here $\{r_n: n \in \mathbb{N}\} \subset \mathbb{N}$ with $r_1 < r_2 < \ldots$ is sublacunary if $\lim_{n \to \infty} (\log r_n)/n = 0$. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for $2$-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.

Citation: Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021175
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