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

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.