Assouad dimension of the Takagi function

For any integer $b\ge 2$ and real series $\{c_n\}$ such that ${\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }{c}_n\mathrm{\mid }<\mathrm{\infty }$, the generalized Takagi function $f_{\mathbf{c},b}(x)$ is defined by $$f_{\mathbf{c},b}(x):=\sum _{n=0}^{\mathrm{\infty }}{c}_n\phi (b^{n}x),x\in [0,1],$$ 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 ${\mathrm{}}_{n\to \mathrm{\infty }}{b}^n\mathrm{\mid }{c}_n\mathrm{\mid }<\mathrm{\infty }$, 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 $b\ge 2$, we define Takagi function $T_{a,b}$ as followed, $$T_{a,b}(x):=\sum _{n=0}^{\mathrm{\infty }}{a}^n\phi (b^{n}x),x\in [0,1]\mathrm{.}$$ Then ${\dim }_A\mathcal{G}{T}_{a,b}=1$ if and only if $0<a\le 1\mathrm{/}b$.