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arXiv · 2601.13296

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

Andreas Rusu·Gabriela Ileana Sebe·Dan Lascu
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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.

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Andreas Rusu, Gabriela Ileana Sebe, Dan Lascu. 2026-01-19. Limit Theorems for $θ$-expansions and the Failure of the Strong Law. https://arxiv.org/abs/2601.13296

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