Measure theoretic properties of large products of consecutive partial quotients

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 $\mathcal{F}_3(\varphi)$.