Assouad dimension of the Takagi function
Lai Jiang
TL;DR
This paper determines the Assouad dimension of graphs of Takagi-type functions. It proves that for the generalized Takagi function $f_{\mathbf{c},b}(x)=\sum_{n=0}^{\infty} c_n\phi(b^n x)$ with $b\ge2$ and $\varlimsup_{n\to\infty} b^n|c_n|<\infty$, the graph has Assouad dimension $\dim_A(\mathcal{G}f_{\mathbf{c},b})=1$, using Lipschitz control of partial sums and a delicate covering argument. It further characterizes the two-parameter Takagi family by showing $\dim_A(\mathcal{G}T_{a,b})=1$ if and only if $0\le a\le 1/b$, implying $\dim_A(\mathcal{G}T_b)=1$ for all $b\ge2$. The analysis hinges on constructing and estimating partial sums $H_n$ and $H_{n,m}$, establishing linearity and Lipschitz properties, and translating these into sharp covering bounds via $\delta$-grid arguments, thereby resolving the Assouad-dimension behavior for a broad class of nowhere differentiable 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 $$ f_{{\mathbf c},b}(x):=\sum_{n=0}^\infty c_nφ(b^n x), \quad x\in [0,1], $$ where $φ(x)=dist(x,\mathbb{Z})$ 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, $$ T_{a,b}(x):=\sum_{n=0}^\infty a^n φ(b^n x), \quad x\in [0,1]. $$ Then $ \dim_A {\mathcal G} T_{a,b}=1 $ if and only if $0<a \leq 1/b$.