The Prime times of twisted Diophantine approximation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915892535754752 |
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| author | Hauke, Manuel |
| author_facet | Hauke, Manuel |
| contents | The seminal work of Kurzweil (1955) provides for any fixed badly approximable $α$ and monotonically decreasing $ψ$ a Khintchine-type statement on the set of the inhomogeneous real parameters $γ$ for which $\lVert n α+ γ\rVert \leq ψ(n)$ has infinitely many integer solutions, and further shows that the assumption of $α$ being badly approximable is necessary. In this article, we generalize Kurzweil's statement to restricting $n \in \mathcal{A}$, where $\mathcal{A} \subseteq \mathbb{N}$ is a set with some multiplicative structure. We show that for badly approximable $α$, the result of Kurzweil extends to a general class of sets $\mathcal{A}$, which allows us to establish the Kurzweil-type result in particular along the primes and along the sums of two squares. Furthermore, we construct non-trivial sets $\mathcal{A}$ where the assumption of $α$ being badly approximable is necessary. In particular, this criterion applies to $\mathcal{A}$ being the set of square-free numbers, providing a novel characterization of the badly approximable numbers. These statements in particular allow for improving the best known bounds for $\lVert n α+ γ\rVert \leq ψ(n)$ for infinitely many $n \in \mathcal{A}$ for fixed badly approximable $α$ and for various sets $\mathcal{A}$ of number-theoretic interest when accepting an exceptional set for $γ$ of Lebesgue measure $0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25291 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Prime times of twisted Diophantine approximation Hauke, Manuel Number Theory Dynamical Systems 11K60 (Primary) 11J54 11J71 11N36 (Secondary) The seminal work of Kurzweil (1955) provides for any fixed badly approximable $α$ and monotonically decreasing $ψ$ a Khintchine-type statement on the set of the inhomogeneous real parameters $γ$ for which $\lVert n α+ γ\rVert \leq ψ(n)$ has infinitely many integer solutions, and further shows that the assumption of $α$ being badly approximable is necessary. In this article, we generalize Kurzweil's statement to restricting $n \in \mathcal{A}$, where $\mathcal{A} \subseteq \mathbb{N}$ is a set with some multiplicative structure. We show that for badly approximable $α$, the result of Kurzweil extends to a general class of sets $\mathcal{A}$, which allows us to establish the Kurzweil-type result in particular along the primes and along the sums of two squares. Furthermore, we construct non-trivial sets $\mathcal{A}$ where the assumption of $α$ being badly approximable is necessary. In particular, this criterion applies to $\mathcal{A}$ being the set of square-free numbers, providing a novel characterization of the badly approximable numbers. These statements in particular allow for improving the best known bounds for $\lVert n α+ γ\rVert \leq ψ(n)$ for infinitely many $n \in \mathcal{A}$ for fixed badly approximable $α$ and for various sets $\mathcal{A}$ of number-theoretic interest when accepting an exceptional set for $γ$ of Lebesgue measure $0$. |
| title | The Prime times of twisted Diophantine approximation |
| topic | Number Theory Dynamical Systems 11K60 (Primary) 11J54 11J71 11N36 (Secondary) |
| url | https://arxiv.org/abs/2603.25291 |