标题： 连分数误差和函数图的Hausdorff维数 显示英文标题

标题： Hausdorff dimension of the graph of the error-sum function of continued fractions

摘要： 我们研究无权重误差和函数$\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$，其中$p_n(x)/q_n(x)$是$x \in \mathbb{R}$的连分数展开的第$n$个渐进分数。 我们证明了$\mathcal{E}$图的豪斯多夫维数正好是$1$。 我们的证明在本质上是数论的，涉及莫比乌斯反演、互质渐进分数分母的求和，以及通过连分数递推关系得出的精确上界。 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 $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$. 显示英文摘要

摘要： 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 $1$. 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 $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x-p_n(x))$.