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

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, L\"u, 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.