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Limit Theorems for $θ$-expansions and the Failure of the Strong Law

Andreas Rusu, Gabriela Ileana Sebe, Dan Lascu

TL;DR

This work extends the metrical theory of continued fractions to $\theta$-expansions by establishing fundamental limit theorems for digit sums under the explicit invariant measure $\gamma_{\theta}$. It proves a Khinchine-type weak law and a Diamond–Vaaler-type result for the sum minus the largest digit, leveraging truncation, precise digit distribution, and the $\psi$-mixing property of the theta-dynamics. The main contribution is a Philipp-style dichotomy: for regular norming sequences $a(n)$, the strong law for $S_{n,\theta}$ fails, with $S_{n,\theta}/a(n)$ either diverging or vanishing according to whether $\sum 1/a(n)$ diverges or converges, respectively. Methodologically, the paper hinges on an explicit Gibbs-Markov structure, exponential decay of correlations, and careful control of variances and covariances of truncated sums, enabling sharp probabilistic limit statements despite non-integrable digit means. This advances our understanding of non-standard ergodic sums in generalized continued fraction systems and generalizes classical results from regular continued fractions to the $\theta$-expansion setting.

Abstract

The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $θ$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the $θ$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties.

Limit Theorems for $θ$-expansions and the Failure of the Strong Law

TL;DR

This work extends the metrical theory of continued fractions to -expansions by establishing fundamental limit theorems for digit sums under the explicit invariant measure . It proves a Khinchine-type weak law and a Diamond–Vaaler-type result for the sum minus the largest digit, leveraging truncation, precise digit distribution, and the -mixing property of the theta-dynamics. The main contribution is a Philipp-style dichotomy: for regular norming sequences , the strong law for fails, with either diverging or vanishing according to whether diverges or converges, respectively. Methodologically, the paper hinges on an explicit Gibbs-Markov structure, exponential decay of correlations, and careful control of variances and covariances of truncated sums, enabling sharp probabilistic limit statements despite non-integrable digit means. This advances our understanding of non-standard ergodic sums in generalized continued fraction systems and generalizes classical results from regular continued fractions to the -expansion setting.

Abstract

The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as -expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the -expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties.
Paper Structure (10 sections, 10 theorems, 96 equations)

This paper contains 10 sections, 10 theorems, 96 equations.

Table of Contents

  1. Introduction and Setup
  2. $\theta$-Expansions
  3. Preliminaries
  4. Distribution of the digits
  5. $\psi$-mixing property
  6. Proof of Theorem \ref{['thm:khinchine']}(Khinchine's Weak Law)
  7. Proof of Theorem \ref{['thm:diamondvaaler']} (Diamond--Vaaler for $\theta$-expansions)
  8. Proof of Theorem \ref{['thm:philipp']} (Failure of the Strong Law)
  9. Case 1: $\sum_{n=1}^\infty \frac{1}{a(n)} = \infty$
  10. Case 2: $\sum_{n=1}^\infty \frac{1}{a(n)} < \infty$

Key Result

Theorem 1.3

Let $\theta \in (0, 1)$ such that $\theta^2 = 1/m$ for some $m \in \mathbb{N}_+$. For every $\varepsilon>0$,

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2: Invariant Measure
  • Theorem 1.3: Khinchine’s Theorem for $\theta$-Expansions
  • Theorem 1.4: Diamond--Vaaler Theorem for $\theta$-Expansions
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Failure of the Strong Law for $\theta$-Expansions
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 13 more