A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914688819789824 |
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| author | Blackman, John Kristensen, Simon Northey, Matthew J. |
| author_facet | Blackman, John Kristensen, Simon Northey, Matthew J. |
| contents | In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $τ(ϕ^ω(a))$ where $ϕ^ω(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|ϕ^n(u)|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09853 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture Blackman, John Kristensen, Simon Northey, Matthew J. Number Theory Combinatorics 11J04 11J61 In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $τ(ϕ^ω(a))$ where $ϕ^ω(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|ϕ^n(u)|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words. |
| title | A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture |
| topic | Number Theory Combinatorics 11J04 11J61 |
| url | https://arxiv.org/abs/2306.09853 |