Hausdorff dimension of Graphs of Limit Functions Generated by Quasi-Linear Functions

TL;DR

The work determines the Hausdorff dimension of graphs of limit functions generated by quasi-linear sequences, showing that for the Rudin–Shapiro abelian-complexity limit $\lambda$, the graph on $(0,1)$ has $\dim_H=\tfrac32$. It then generalizes to the graphs of limit functions $\lambda_{\mathbf{s}}$ arising from quasi-linear sequences, establishing $\dim_H \mathrm{Gr}_{u,v}(\lambda_{\mathbf{s}})=2-\alpha$ under conditions $\alpha(\mathbf{s})=\alpha$, $\beta(\mathbf{s})=0$ and a syndetic-type growth constraint on $a_{\mathbf{s}}$. The proofs combine Lipschitz embeddings between the graphs, and mass-distribution arguments on the auxiliary function $a_{\mathbf{s}}(x)$ to derive sharp lower bounds, complementing existing box-dimension results. The examples with the Thue–Morse and Rudin–Shapiro sequences demonstrate sharpness and illuminate the necessity of the hypotheses.

Abstract

The limit functions generated by quasi-linear functions or sequences (including the sum of the Rudin-Shapiro sequence as an example) are continuous but almost everywhere non-differentiable functions. Their graphs are fractal curves. In 2017 and 2020, Chen, Lü, Wen and the first author studied the box dimension of the graphs of the limit functions. In this paper, we focus on the Hausdorff dimension of the graphs of such limit functions. We first prove that the Hausdorff dimension of the graph of the limit function generated by the abelian complexity of the Rudin-Shapiro sequence is $\frac{3}{2}$. Then we extend the result to the graphs of limit functions generated by quasi-linear functions.

Theorems & Definitions (30)

• Theorem 1
• Definition 2: Quasi-linear functions
• Theorem 3
• Lemma 4: Proposition 2 in LCWW17
• Remark 5
• Lemma 6: cf. LCWW17, Theorem 3
• Lemma 7
• proof
• Lemma 8: Mass Distribution Principle
• Lemma 9