Abstract: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 $\varphi:\mathbb{N}\to\mathbb{R}_{\geq 2}$, we determine the Lebesgue measure of the set $\mathcal{F}_{\ell}(\varphi)$ of irrational numbers $x$ whose regular continued fraction $x~=~[a_1(x),a_2(x),\ldots]$ is such that, for infinitely many $n\in\mathbb{N}$, there are two numbers $1\leq j<k \leq n$ satisfying \[ a_{k}(x)\cdots a_{k+\ell-1}(x) \geq \varphi(n), \; a_{j}(x)\cdots a_{j+\ell-1}(x) \geq \varphi(n). \] This result generalizes previous work by Tan and Zhou (Nonlinearity, 2024). A consequence of our result is that the strong law of large numbers for products of $\ell$ consecutive partial quotients is impossible even if the block with the largest product is removed. We also compute the Hausdorff dimension of $\mathcal{F}_3(\varphi)$.
From: Gerardo González Robert Mr. [view email]
[v1]
Fri, 17 May 2024 04:39:27 UTC (24 KB)
[v2]
Tue, 22 Jul 2025 23:45:04 UTC (25 KB)
[v3]
Tue, 16 Jun 2026 20:34:49 UTC (31 KB)
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