Assouad dimension of the Takagi function
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909537033781248 |
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| author | Jiang, Lai |
| author_facet | Jiang, Lai |
| contents | 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 $0<a \leq 1/b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Assouad dimension of the Takagi function Jiang, Lai Classical Analysis and ODEs 28A80 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 $0<a \leq 1/b$. |
| title | Assouad dimension of the Takagi function |
| topic | Classical Analysis and ODEs 28A80 |
| url | https://arxiv.org/abs/2502.01140 |