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

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.

Improved bounds for $({\alpha }_n)$

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Cited by in corpus (1)

• Quenched and annealed equilibrium states for random Ruelle expanding maps and applications