Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals

TL;DR

This paper addresses the fractal structure of irrationals with irrationality exponent $\mu(x)>2$ by analyzing their intersections with backward continued fraction (BCF) fractals generated by a parabolic iterated function system. It proves that for any finite BCF alphabet $\mathcal B$ containing $2$, the Hausdorff dimension of $G(\alpha)\cap F_{\mathcal B}$ tends to $\dim_H F_{\mathcal B}$ as $\alpha\downarrow 2$, and derives the corollaries that $\dim_H\{x:\mu(x)>2\text{ with bounded BCF} \}=1$ while such a phenomenon does not occur for regular continued fractions. The method hinges on a dynamical-systems approach: (i) representing $F_{\mathcal B}$ as a parabolic IFS, (ii) extracting a no-parabolic finite IFS with a seed of near-optimal dimension, (iii) inserting long blocks of the digit $2$ to enforce $G(\alpha)$-growth, and (iv) transferring dimension estimates back to the BCF limit set via distortion and Lipschitz control. The paper further analyzes the irrationality exponent for BCF, showing that in the bounded-$2$-block regime a BCF expression for $\mu(x)$ matches a corresponding limsup, while outside this regime the relation can fail on a set of dimension $1$, underscoring a qualitative difference between BCF and RCF descriptions. Overall, the work reveals that BCF-based fractals retain full dimensional intersections with Jarník sets and yields sharp contrasts with RCF-based arithmetic properties.

Abstract

The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal structure. We show that this set intersects the limit set of any parabolic iterated function system arising from the backward continued fraction in a set of full Hausdorff dimension. As a corollary, we show that the set of irrationals whose irrationality exponents are strictly bigger than $2$ and whose backward continued fraction expansions have bounded partial quotients is of Hausdorff dimension $1$. This is a sharp contrast to the fact that there exists no irrational whose irrationality exponent is strictly greater than $2$ and whose regular continued fraction expansion has bounded partial quotients.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2: G41
• Remark 2.3
• Proposition 2.4
• proof