We study the unweighted error-sum function $ \mathcal{E}(x) : = \sum_{n \geq 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 \geq 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 $.
Citation: Min Woong Ahn. Hausdorff dimension of the graph of the unweighted error-sum function of continued fractions[J]. AIMS Mathematics, 2026, 11(10): 34081-34106. doi: 10.3934/math.20261329
We study the unweighted error-sum function $ \mathcal{E}(x) : = \sum_{n \geq 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 \geq 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 $.
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Min Woong Ahn. Hausdorff dimension of the graph of the unweighted error-sum function of continued fractions[J]. AIMS Mathematics, 2026, 11(10): 34081-34106. doi: 10.3934/math.20261329
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