An improvement of the lower bound of the number of integers in Littlewood's conjecture
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866910415458402304 |
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| author | Usuki, Shunsuke |
| author_facet | Usuki, Shunsuke |
| contents | In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<γ<1$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ(\log N)^2/(\log\log N)^2$ up to a universal constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05027 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improvement of the lower bound of the number of integers in Littlewood's conjecture Usuki, Shunsuke Number Theory Dynamical Systems 11J13, 37A17 In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<γ<1$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ(\log N)^2/(\log\log N)^2$ up to a universal constant. |
| title | An improvement of the lower bound of the number of integers in Littlewood's conjecture |
| topic | Number Theory Dynamical Systems 11J13, 37A17 |
| url | https://arxiv.org/abs/2401.05027 |