On the Hausdorff dimension of graph of random vector-valued Weierstrass function
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908966668206080 |
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| author | Luo, Jun Jason Zhang, Zi-Rui |
| author_facet | Luo, Jun Jason Zhang, Zi-Rui |
| contents | Let $Θ=\{θ_n\}, Λ=\{λ_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{Θ,Λ}(t)=
\left(
\sum_{n=0}^{\infty} b^{-βn}\cos\bigl(2π(b^n t+θ_n)\bigr),\
\sum_{n=0}^{\infty} b^{-βn}\sin\bigl(2π(b^n t+λ_n)\bigr)
\right),\quad t\in[0,1],
\] where $b>1, β\in (0,1/2)$. We prove that, with probability one, the Hausdorff dimension of the graph of this function is
\[
\dim_H G(f_{Θ,Λ})=3-2β,
\] extending a result of Hunt in 1998. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff dimension of graph of random vector-valued Weierstrass function Luo, Jun Jason Zhang, Zi-Rui Classical Analysis and ODEs Probability 28A78 Let $Θ=\{θ_n\}, Λ=\{λ_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{Θ,Λ}(t)= \left( \sum_{n=0}^{\infty} b^{-βn}\cos\bigl(2π(b^n t+θ_n)\bigr),\ \sum_{n=0}^{\infty} b^{-βn}\sin\bigl(2π(b^n t+λ_n)\bigr) \right),\quad t\in[0,1], \] where $b>1, β\in (0,1/2)$. We prove that, with probability one, the Hausdorff dimension of the graph of this function is \[ \dim_H G(f_{Θ,Λ})=3-2β, \] extending a result of Hunt in 1998. |
| title | On the Hausdorff dimension of graph of random vector-valued Weierstrass function |
| topic | Classical Analysis and ODEs Probability 28A78 |
| url | https://arxiv.org/abs/2604.13913 |