On multiplicative recurrence along linear patterns
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908499654475776 |
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| author | Charamaras, Dimitrios Mountakis, Andreas Tsinas, Konstantinos |
| author_facet | Charamaras, Dimitrios Mountakis, Andreas Tsinas, Konstantinos |
| contents | In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N}, \times)$ and provided some sufficient conditions for sets of the form $S=\{(an+b)/(cn+d) \colon n \in \mathbb{N} \}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values on the unit circle, we have that $\liminf_{n \to \infty} |f(an+b)-f(cn+d)|=0.$ In this article, we fully characterize the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. In addition, we prove that, under the same conditions on $(a,b,c,d)$, the set $S$ is a set of recurrence for finitely generated actions of $(\mathbb{N}, \times)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On multiplicative recurrence along linear patterns Charamaras, Dimitrios Mountakis, Andreas Tsinas, Konstantinos Number Theory Dynamical Systems 11N37 (Primary) 37A44 (Secondary) In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N}, \times)$ and provided some sufficient conditions for sets of the form $S=\{(an+b)/(cn+d) \colon n \in \mathbb{N} \}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values on the unit circle, we have that $\liminf_{n \to \infty} |f(an+b)-f(cn+d)|=0.$ In this article, we fully characterize the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. In addition, we prove that, under the same conditions on $(a,b,c,d)$, the set $S$ is a set of recurrence for finitely generated actions of $(\mathbb{N}, \times)$. |
| title | On multiplicative recurrence along linear patterns |
| topic | Number Theory Dynamical Systems 11N37 (Primary) 37A44 (Secondary) |
| url | https://arxiv.org/abs/2412.03504 |