2021/08/04 by John T. Griesmer, Griesmer, John T. · 1 citation
Mathematics · #37B20 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2108.02190
openalex publication_date 2021/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊂ G is an I0 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 \2n-2m : n,m∈ \mathbb N\ which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let \mathbb Zω 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 Fpω be the direct sum of countably many copies of \mathbb Z/p\mathbb Z with basis (\mathbf ei)i∈ \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 Fpω determined by \∑i∈ F\mathbf ei:F∈ \mathcal F\ has infinite chromatic number, then every subset of \mathbb Fpω which is a set of Bohr recurrence is a set of topological recurrence.