M\"obius inversion and coprime summation for error-sum functions of continued fractions

TL;DR

This paper investigates the Hausdorff dimension of the graph of an error-sum function related to continued fractions, proving it is exactly 1 using number-theoretic methods involving M"obius inversion and coprime summations.

Contribution

It provides a novel number-theoretic proof that the Hausdorff dimension of the error-sum function's graph is exactly 1, and rederives bounds for related functions.

Findings

Hausdorff dimension of the error-sum function's graph is exactly 1

Number-theoretic methods are used in the proof

Re-derivation of the 3/2 upper bound for the relative error-sum function

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 M\"obius 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 $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.