Sublacunary sets and interpolation sets for nilsequences

<p style='text-indent:20px;'>A set <inline-formula><tex-math id="M1">\begin{document}$ E \subset \mathbb{N} $\end{document}</tex-math></inline-formula> is an interpolation set for nilsequences if every bounded function on <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> can be extended to a nilsequence on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{N} $\end{document}</tex-math></inline-formula>. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here <inline-formula><tex-math id="M4">\begin{document}$ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ r_1 &lt; r_2 &lt; \ldots $\end{document}</tex-math></inline-formula> is <i>sublacunary</i> if <inline-formula><tex-math id="M6">\begin{document}$ \lim_{n \to \infty} (\log r_n)/n = 0 $\end{document}</tex-math></inline-formula>. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for <inline-formula><tex-math id="M7">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.</p>