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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.19582 |
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| _version_ | 1866914580277493760 |
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| author | Allen, Demi Hauke-Treuer, Manuel Ramírez, Felipe A. |
| author_facet | Allen, Demi Hauke-Treuer, Manuel Ramírez, Felipe A. |
| contents | We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The inhomogeneous Khintchine Theorem in dimension two Allen, Demi Hauke-Treuer, Manuel Ramírez, Felipe A. Number Theory We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart. |
| title | The inhomogeneous Khintchine Theorem in dimension two |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.19582 |