On the fractal dimension and structure of exceptional sets in $\vartheta$-expansions

Abstract

This paper studies the metrical theory of $\vartheta$-expansions, a generalization of regular continued fractions. We focus on the Hausdorff dimension of two classical types of exceptional sets. First, we extend Jarník’s results on the dimension of sets of numbers with bounded partial quotients to the $\vartheta$-expansion setting, obtaining new bounds that improve upon the classical ones in the special case of regular continued fractions. Second, if ${\mathcal{F}}_n(x)$ is the largest partial quotient of the $\vartheta$-expansion, we prove that for all $\beta \ge 0$, the set of numbers $x$ for which $({\mathcal{F}}_n(x)\log \log n)/n$ converges to $\beta$ has full Hausdorff dimension. This result complements a previous almost everywhere law and generalizes the work of Philipp (1975/76), Okano (2002), Wu and Xu (2009) to $\vartheta$-expansions.