Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Manuel Stadlbauer, Xuan Zhang

Abstract

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $α_n$, where $(α_n)$ is a sequence such that $\sum 1/α_n$ 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.

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 -th partial quotient is bigger than , where is a sequence such that is finite. This set is shown to have Hausdorff dimension 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.

Paper Structure

This paper contains 9 sections, 12 theorems, 92 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Non-stationary shift spaces and decay of correlations
  3. An almost sure invariance principle
  4. The law of the iterated logarithm
  5. The geometric potential
  6. Expected value and variance of the logarithm.
  7. Law of the iterated logarithm
  8. Hausdorff dimension and measure
  9. Proof of Theorem \ref{['theo:easy_bounds']}

Key Result

Theorem 1.1

Assume that $\sum_n 1/\alpha_n < \infty$ and let $\gamma_n$ be given by $\gamma_n \alpha_n^{\gamma_n}=1$. Then there exists a probability measure $\nu$ on $X_\alpha$ for which the following holds almost surely. Furthermore, if $n\ll \alpha_n \ll \lambda^{n}$ or $\lambda^{n} \ll \alpha_n \ll n^{n(1-\epsilon)}$ for some $\lambda >1$ and $\epsilon >0$, then $\nu$ is absolutely continuous with resp

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1: CunyMerlevede:2015
  • Theorem 3.2
  • ...and 13 more