Hausdorff dimension estimates for Sudler products with positive lower bound
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914796273664000 |
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| author | Gayfulin, Dmitry Hauke, Manuel |
| author_facet | Gayfulin, Dmitry Hauke, Manuel |
| contents | Given an irrational number $α$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(α) = \prod_{r=1}^N 2\lvert \sin πr α\rvert$. We show that $\liminf_{N \to \infty} P_N(α) >0$ and $\limsup_{N \to \infty} P_N(α)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $α$ that satisfy $\limsup_{N \to \infty} P_N(α)/N < \infty,\liminf_{N \to \infty} P_N(α) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of such $α$ is invariant under the Gauss map $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_06548 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension estimates for Sudler products with positive lower bound Gayfulin, Dmitry Hauke, Manuel Number Theory 11J70, 26D05 Given an irrational number $α$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(α) = \prod_{r=1}^N 2\lvert \sin πr α\rvert$. We show that $\liminf_{N \to \infty} P_N(α) >0$ and $\limsup_{N \to \infty} P_N(α)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $α$ that satisfy $\limsup_{N \to \infty} P_N(α)/N < \infty,\liminf_{N \to \infty} P_N(α) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of such $α$ is invariant under the Gauss map $T$. |
| title | Hausdorff dimension estimates for Sudler products with positive lower bound |
| topic | Number Theory 11J70, 26D05 |
| url | https://arxiv.org/abs/2312.06548 |