Estimating lower limit in the $p$-adic Littlewood conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908414857183232 |
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| author | Badziahin, Dmitry |
| author_facet | Badziahin, Dmitry |
| contents | We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<ε$ for all real $x$, small primes $p$ and relatively small $ε$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower limit is equal to 0 for all $x\in\mathbb{R}$. In particular, the result is established for $p=2$ with $ε=1/25$. For $3\le p\le 29$, the upper bounds for $ε$ vary, but they are always at most $1/10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimating lower limit in the $p$-adic Littlewood conjecture Badziahin, Dmitry Number Theory 11J13 We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<ε$ for all real $x$, small primes $p$ and relatively small $ε$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower limit is equal to 0 for all $x\in\mathbb{R}$. In particular, the result is established for $p=2$ with $ε=1/25$. For $3\le p\le 29$, the upper bounds for $ε$ vary, but they are always at most $1/10$. |
| title | Estimating lower limit in the $p$-adic Littlewood conjecture |
| topic | Number Theory 11J13 |
| url | https://arxiv.org/abs/2506.16860 |