Hausdorff dimension of the graph of the unweighted error-sum function of continued fractions

Abstract

We study the unweighted error-sum function $\mathcal{E}(x):={\sum }_{n\ge 0}(x-p_n(x)/q_n(x))$, where $p_n(x)/q_n(x)$ denotes the $n$th convergent of the continued fraction expansion of $x\in \mathbb{R}$, and we prove that the graph of $\mathcal{E}$ has Hausdorff dimension $1$. The key ingredient is an exact formula for the oscillation of $\mathcal{E}$ on fundamental intervals, which we obtain in a symbolic space of continued fraction sequences containing non-realizable sequences. Using the same framework, we also give a new derivation of the upper bound $3/2$ of Ridley and Petruska (2000) for the Hausdorff dimension of the graph of the relative error-sum function $P(x):={\sum }_{n\ge 0}(q_n(x)x-p_n(x))$. Moreover, we present an alternative argument based on Möbius inversion and summation over coprime convergent denominators, and we explain why the square coverings used for $P$ cannot yield an upper bound smaller than $3/2$.

arXiv comment29 pages, typos corrected