Measure theoretic properties of large products of consecutive partial quotients (2405.10538v1)

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 $\varphi:\mathbb{N}\to\mathbb{R}{>0}$ and $\ell\in \mathbb{N}$, we determine the Lebesgue measure and Hausdorff dimension 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). ] One of the consequences of the results 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.