Sublacunary sets and interpolation sets for nilsequences

Abstract

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