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On the dimension of the graph of the classical Weierstrass function
- Speaker(s)
- Krzysztof Barański
- Affiliation
- Uniwersytet Warszawski
- Date
- Nov. 8, 2013, 10:15 a.m.
- Room
- room 5840
- Seminar
- Seminar of Dynamical Systems Group
We examine dimension of the graph of the famous Weierstrass non-differentiable function
\[
W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x)
\]
for an integer $b$ larger than $1$ and $1/b < \lambda < 1$. We prove that for every $b$ there exists (explicitly given) $\lambda_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph is equal to $D = 2+\frac{\log\lambda}{\log b}$ for every $\lambda\in(\lambda_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. The results solve partially a well-known thirty-year-old conjecture.
This is a joint work with Balazs Barany and Julia Romanowska.
\[
W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x)
\]
for an integer $b$ larger than $1$ and $1/b < \lambda < 1$. We prove that for every $b$ there exists (explicitly given) $\lambda_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph is equal to $D = 2+\frac{\log\lambda}{\log b}$ for every $\lambda\in(\lambda_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. The results solve partially a well-known thirty-year-old conjecture.
This is a joint work with Balazs Barany and Julia Romanowska.
