Möbius inversion and coprime summation for error-sum functions of continued fractions

TL;DR

The paper investigates the fractal geometry of error-sum functions associated with continued fraction expansions. It develops a number-theoretic framework using Möbius inversion and coprime-denominator summations to control the graphs’ Hausdorff dimensions via explicit coverings tied to CF cylinder sets. The authors prove the unweighted error-sum graph has Hausdorff dimension exactly $1$ and provide a new derivation of the known upper bound $3/2$ for the relative error-sum graph, highlighting a path toward resolving the open problem of the exact dimension of the latter. The approach blends continued fraction recurrence analysis with fractal-geometry techniques, offering new perspectives and potential directions for future work on $P$.

Abstract

We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff dimension of the graph of $\mathcal{E}$ is exactly equal to $1$. Our proof is number-theoretic in nature and involves Möbius inversion, summation over coprime convergent denominators, and precise upper bounds derived via continued fraction recurrence relations. As a supplementary result, we rederive the known upper bound of $3/2$ for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2: RP00
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3: IK02
• Proposition 2.4: See Ahn25HW08IK02Khi97RP00
• Proposition 2.5: Ahn25
• Proposition 2.6