Measure theoretic properties of large products of consecutive partial quotients

TL;DR

This paper investigates the measure and dimension of sets of irrationals with large products of consecutive partial quotients in their continued fractions, revealing limitations of the strong law of large numbers for these products.

Contribution

It characterizes the Lebesgue measure and Hausdorff dimension of sets defined by large products of consecutive partial quotients, extending understanding of their measure-theoretic properties.

Findings

Determines measure and dimension of sets with large partial quotient products.

Shows the strong law of large numbers fails for these products even after removing the largest block.

Provides conditions on functions for measure and dimension results.

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 $\varphi:\mathbb{N}\to [2,\infty)$, 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 \[ a_{k}(x)a_{k+1}(x)a_{k+2}(x) \geq \varphi(n), \; a_{j}(x)a_{j+1}(x)a_{j+2}(x) \geq \varphi(n). \] 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.