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

TL;DR

This paper establishes metrical theorems for $ heta$-expansions, including laws of large numbers and a general failure of the strong law, extending classical results using invariant measures and mixing properties.

Contribution

It introduces new limit theorems for $ heta$-expansions, notably demonstrating the failure of the strong law with no regular norming sequence, extending Philipp's classical theorem.

Findings

Proved Khinchine's Weak Law for $ heta$-digits.

Established the Diamond-Vaaler Strong Law for sum of digits minus the largest.

Showed the failure of the strong law for $ heta$-expansions with no finite, non-zero limit.

Abstract

The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $\theta$-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 $\theta$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties.