Geodesic
Separating Bohr denseness from measurable recurrence
Discrete analysis (2021) Cet article a éte moissonné depuis la source Scholastica

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We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb Z$ is dense in the Bohr topology of $\mathbb Z$, then there is a set $S'\subseteq S$ such that $S'$ is dense in the Bohr topology of $\mathbb Z$ and for all $m\in \mathbb Z,$ the set $(S'-m)\setminus \{0\}$ is not a set of measurable recurrence.
Publié le : 
@article{DAS_2021_a18,
     author = {John T. Griesmer},
     title = {Separating {Bohr} denseness from measurable recurrence},
     journal = {Discrete analysis},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2021_a18/}
}
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LA  - en
ID  - DAS_2021_a18
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John T. Griesmer. Separating Bohr denseness from measurable recurrence. Discrete analysis (2021). http://geodesic.mathdoc.fr/item/DAS_2021_a18/