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