Inhomogeneous Khintchine-Groshev theorem without monotonicity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912443495612416 |
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| author | Kim, Seongmin |
| author_facet | Kim, Seongmin |
| contents | The Khintchine-Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of $ψ$-approximable numbers, given a monotonic function $ψ$. Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem for cases with $nm\geq3$ and conjectured that it also holds for $nm=2$. In this paper, we prove this conjecture in the case of $(n,m)=(2,1)$. We also prove it for the case of $(n,m)=(1,2)$ with a rational inhomogeneous parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous Khintchine-Groshev theorem without monotonicity Kim, Seongmin Number Theory 11J20, 11K60, 11J83 The Khintchine-Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of $ψ$-approximable numbers, given a monotonic function $ψ$. Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem for cases with $nm\geq3$ and conjectured that it also holds for $nm=2$. In this paper, we prove this conjecture in the case of $(n,m)=(2,1)$. We also prove it for the case of $(n,m)=(1,2)$ with a rational inhomogeneous parameter. |
| title | Inhomogeneous Khintchine-Groshev theorem without monotonicity |
| topic | Number Theory 11J20, 11K60, 11J83 |
| url | https://arxiv.org/abs/2411.07932 |