Interpolation sets and nilsequences
Volume 162 / 2020
Colloquium Mathematicum 162 (2020), 181-199
MSC: Primary 37A45; Secondary 11B30.
DOI: 10.4064/cm7937-9-2019
Published online: 27 April 2020
Abstract
To answer a question of Frantzikinakis, we study a class of subsets of $\mathbb {N}$, called interpolation sets, on which every bounded sequence can be extended to an almost periodic sequence on $\mathbb {N}$. It has been proved by Strzelecki that lacunary sets are interpolation sets. We prove that sets that are denser than all lacunary sets cannot be interpolation sets. We also extend the notion of interpolation sets to nilsequences and show that the analogue to Frantzikinakis’s question for arbitrary sequences has a negative answer.
