Sarnak's Program for Erdős Sieves. Part I: Topological Dynamics and Light Tails
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908855874617344 |
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| author | Araújo, Francisco |
| author_facet | Araújo, Francisco |
| contents | This paper is the first part of a two-part article where we generalize Sarnak's program to sets where we remove congruence classes modulo some infinite set $\mathcal{B}$ of ideals of an étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős sieves. We define some light tail conditions on a sieve $R$, and show how these are related to the genericity under the Mirsky measure of the set of $R-$free numbers, which are the algebraic integers of $K$ not contained in any of the congruence classes in $R$. We also show that Erdős $\mathcal{B}-$free systems in any étale $\mathbb{Q}-$algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erdős $\mathcal{B}-$free systems over any étale $\mathbb{Q}-$algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_24031 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sarnak's Program for Erdős Sieves. Part I: Topological Dynamics and Light Tails Araújo, Francisco Dynamical Systems Number Theory This paper is the first part of a two-part article where we generalize Sarnak's program to sets where we remove congruence classes modulo some infinite set $\mathcal{B}$ of ideals of an étale $\mathbb{Q}-$algebra $K$, which we denote by Erdős sieves. We define some light tail conditions on a sieve $R$, and show how these are related to the genericity under the Mirsky measure of the set of $R-$free numbers, which are the algebraic integers of $K$ not contained in any of the congruence classes in $R$. We also show that Erdős $\mathcal{B}-$free systems in any étale $\mathbb{Q}-$algebra satisfy these light tail conditions, so our results generalize Sarnak's program to Erdős $\mathcal{B}-$free systems over any étale $\mathbb{Q}-$algebra. |
| title | Sarnak's Program for Erdős Sieves. Part I: Topological Dynamics and Light Tails |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2602.24031 |