Measure theoretic properties of large products of consecutive partial quotients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913954149695488 |
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| author | Brown-Sarre, Adam Robert, Gerardo González Hussain, Mumtaz |
| author_facet | Brown-Sarre, Adam Robert, Gerardo González Hussain, Mumtaz |
| contents | The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $φ:\mathbb{N}\to [2,\infty)$, we determine the Lebesgue measure and Hausdorff dimension of the set $\mathcal{F}_{\ell}(\vphi)$ of irrational numbers $x$ whose regular continued fraction $x~=~[a_1(x),a_2(x),\ldots]$ is such that for infinitely many $n\in\Na$ there are two numbers $1\leq j<k \leq n$ satisfying \[ a_{k}(x)a_{k+1}(x)a_{k+2}(x) \geq φ(n), \; a_{j}(x)a_{j+1}(x)a_{j+2}(x) \geq φ(n). \] One of the consequences of the results is that the strong law of large numbers for products of $3$ consecutive partial quotients is impossible even if the block with the largest product is removed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure theoretic properties of large products of consecutive partial quotients Brown-Sarre, Adam Robert, Gerardo González Hussain, Mumtaz Number Theory 11K55 (Primary) 11J83, 28A80 (Secondary) The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $φ:\mathbb{N}\to [2,\infty)$, we determine the Lebesgue measure and Hausdorff dimension of the set $\mathcal{F}_{\ell}(\vphi)$ of irrational numbers $x$ whose regular continued fraction $x~=~[a_1(x),a_2(x),\ldots]$ is such that for infinitely many $n\in\Na$ there are two numbers $1\leq j<k \leq n$ satisfying \[ a_{k}(x)a_{k+1}(x)a_{k+2}(x) \geq φ(n), \; a_{j}(x)a_{j+1}(x)a_{j+2}(x) \geq φ(n). \] One of the consequences of the results is that the strong law of large numbers for products of $3$ consecutive partial quotients is impossible even if the block with the largest product is removed. |
| title | Measure theoretic properties of large products of consecutive partial quotients |
| topic | Number Theory 11K55 (Primary) 11J83, 28A80 (Secondary) |
| url | https://arxiv.org/abs/2405.10538 |