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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.24034 |
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| _version_ | 1866910035274104832 |
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| author | Araújo, Francisco |
| author_facet | Araújo, Francisco |
| contents | This paper is the second 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. Given a sieve $R$ we define the set $\mathcal{F}_R$ of algebraic integers in $K$ not contained in any of the congruence classes of $R$. We associate to each sieve two measure-theoretical dynamical systems $X_R$ (the orbit closure of $\mathcal{F}_R$) and $Ω_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $Ω_R$ is isomorphic to an ergodic rotation of a compact abelian group, and compute its spectrum. As applications we show results about infinite sumsets in the integers, investigate the case where $\mathcal{F}_R$ is the squarefree values of some polynomial, and show a prime number theorem for $R-$free numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_24034 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sarnak's Program for Erdős Sieves. Part II: Measure Systems and Applications Araújo, Francisco Dynamical Systems Number Theory This paper is the second 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. Given a sieve $R$ we define the set $\mathcal{F}_R$ of algebraic integers in $K$ not contained in any of the congruence classes of $R$. We associate to each sieve two measure-theoretical dynamical systems $X_R$ (the orbit closure of $\mathcal{F}_R$) and $Ω_R$ (the set of $R-$admissible sets) and show how they are related. We show that the system associated to $Ω_R$ is isomorphic to an ergodic rotation of a compact abelian group, and compute its spectrum. As applications we show results about infinite sumsets in the integers, investigate the case where $\mathcal{F}_R$ is the squarefree values of some polynomial, and show a prime number theorem for $R-$free numbers. |
| title | Sarnak's Program for Erdős Sieves. Part II: Measure Systems and Applications |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2602.24034 |