Special cases and equivalent forms of Katznelson's problem on recurrence

TL;DR

The paper explores special cases and equivalent forms of Katznelson's recurrence problem, establishing conditions under which Bohr recurrence sets are also topological recurrence sets in various algebraic structures.

Contribution

It provides new results linking Bohr and topological recurrence in countable abelian groups, including specific cases involving $I_0$ sets and hypergraph chromatic properties.

Findings

Sets of Bohr recurrence in $I_0$ sets are also sets of topological recurrence.

If certain hypergraph chromatic conditions hold, Bohr recurrence sets in $_p^$ are also topological recurrence.

Implications for countable abelian groups and their subsets regarding recurrence properties.

Abstract

We make three observations regarding a question popularized by Katznelson: is every subset of $\mathbb Z$ which is a set of Bohr recurrence is also a set of topological recurrence? (i) If $G$ is a countable abelian group and $E\subset G$ is an $I_0$ set, then every subset of $E-E$ which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of $\{2^n-2^m : n,m\in \mathbb N\}$ which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let $\mathbb Z^{\omega}$ be the direct sum of countably many copies of $\mathbb Z$ with standard basis $E$. If every subset of $(E-E)-(E-E)$ which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime $p$ and let $\mathbb F_p^\omega$ be the direct sum of countably many copies of $\mathbb Z/p\mathbb Z$ with basis $(\mathbf e_i)_{i\in \mathbb N}$. If for every $p$-uniform hypergraph with vertex set $\mathbb N$ and edge set $\mathcal F$ having infinite chromatic number, the Cayley graph on $\mathbb F_p^\omega$ determined by $\{\sum_{i\in F}\mathbf e_i:F\in \mathcal F\}$ has infinite chromatic number, then every subset of $\mathbb F_p^\omega$ which is a set of Bohr recurrence is a set of topological recurrence.