# Interpolation sets and nilsequences

## TL;DR

This paper investigates interpolation sets in number theory, showing that denser sets than lacunary ones are not interpolation sets, and extends the concept to nilsequences, providing counterexamples to a related question.

## Contribution

It introduces the concept of interpolation sets for sequences and extends it to nilsequences, demonstrating limitations and counterexamples for dense sets.

## Key findings

- Lacunary sets are interpolation sets.
- Denser sets than lacunary are not interpolation sets.
- The analogue of Frantzikinakis' question for nilsequences is false.

## Abstract

To give positive answer to 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 $\mathbb{N}$. Strzelecki has proved 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' question for arbitrary sequences is false.