Sublacunary sets and interpolation sets for nilsequences

TL;DR

This paper investigates the properties of sublacunary sets as interpolation sets for nilsequences, establishing their limitations and introducing new classes of such sets, including examples that distinguish between nilsequences and Bohr almost periodic sequences.

Contribution

It proves that sublacunary sets are not interpolation sets for nilsequences and introduces new classes of interpolation sets, including examples that separate nilsequences from Bohr almost periodic sequences.

Findings

Sublacunary sets are not interpolation sets for nilsequences.

Union of an interpolation set and a finite set remains an interpolation set.

Constructs a new example of an interpolation set for 2-step nilsequences not being one for Bohr sequences.

Abstract

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. 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 the result, obtain a new example of interpolation set for $2$-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.