Sharp Hausdorff Dimension Bounds for Sets with Bounded and Growing Digits in $N$-expansions

Abstract

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we obtain improved Jarn\'ik-type bounds that generalize and refine classical results for regular continued fractions, with explicit dependence on $N$ (Theorem~1.1). For sets with digits that grow without bound, we obtain precise asymptotics that extend Good's theorems to $N$-expansions, proving in particular that the set of numbers whose digits tend to infinity has Hausdorff dimension exactly $1/2$, and that the dimension of sets with uniformly large digits approaches $1/2$ as the lower bound increases, with explicit logarithmic decay (Theorem~1.2). The results reveal how the parameter $N$ influences the dimensional properties of these exceptional sets. Our methods combine careful estimates of fundamental interval lengths with optimized covering arguments for upper bounds and Cantor set constructions via mass distribution principles for lower bounds.