On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients

TL;DR

This paper proves a law of the iterated logarithm for real numbers with constrained continued fraction partial quotients, revealing their Hausdorff dimension and measure properties using advanced probabilistic and dynamical systems techniques.

Contribution

It establishes a new LIL for continued fractions with sequential restrictions and links the measure to Hausdorff measure via a strong invariance principle.

Findings

Hausdorff dimension of the set is 1/2 in many cases

Measure in LIL is absolutely continuous to Hausdorff measure

Provides a strong invariance principle for unbounded observables

Abstract

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system.