Assouad dimension of the Takagi function

TL;DR

This paper investigates the Assouad dimension of the graphs of generalized Takagi functions, establishing that under certain conditions, their graphs have Assouad dimension equal to one, indicating a specific fractal complexity.

Contribution

The paper proves that the Assouad dimension of the graph of generalized Takagi functions is one when a certain boundedness condition on the coefficients holds, and characterizes this dimension for a class of Takagi functions.

Findings

Assouad dimension of the graph is 1 under boundedness condition.

For the specific Takagi functions T_{a,b}, the dimension is 1 if and only if 0 < a ≤ 1/b.

The results provide a precise fractal dimension characterization for these functions' graphs.

Abstract

For any integer $b\geq2$ and real series $\{c_n\}$ such that $\sum_{n=0}^\infty|c_n|<\infty$, the generalized Takagi function $f_{{\mathbf c},b}(x)$ is defined by $$ f_{{\mathbf c},b}(x):=\sum_{n=0}^\infty c_n\phi(b^n x), \quad x\in [0,1], $$ where $\phi(x)=dist(x,\mathbb{Z})$ is the distance from $x$ to the nearest integer. The collection of functions with the form are called the Takagi class. In this paper, we show that in the case that $\varlimsup_{n \to \infty} b^n |c_n|<\infty$, the Assouad dimension of the graph ${\mathcal G} f_{{\mathbf c},b}=\{(x,f_{{\mathbf c},b}(x)):x\in[0,1]\}$ for the generalized Takagi function $f_{{\mathbf c},b}(x)$ is equal to one, that is, $$ \dim_A {\mathcal G} f_{{\mathbf c},b}=1. $$ In particular, for each $0<a<1$ and integer $b \geq 2$, we define Takagi function $T_{a,b}$ as followed, $$ T_{a,b}(x):=\sum_{n=0}^\infty a^n \phi(b^n x), \quad x\in [0,1]. $$ Then $ \dim_A {\mathcal G} T_{a,b}=1 $ if and only if $0<a \leq 1/b$.