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Zbl 0603.28003
Mauldin, R.Daniel; Williams, S.C.
On the Hausdorff dimension of some graphs. (English)
[J] Trans. Am. Math. Soc. 298, 793-803 (1986). ISSN 0002-9947; ISSN 1088-6850/e

Consider the functions $$ W\sb b(x)=\sum\sp{\infty}\sb{n=-\infty}b\sp{- \alpha n}[\Phi (b\sp nx+\theta\sb n)-\Phi (\theta\sb n)], $$ where $b>1$, $0<\alpha <1$, each $\theta\sb n$ is an arbitrary number, and $\Phi$ has period one. We show that there is a constant $C>0$ such that if b is large enough, then the Hausdorff dimension of the graph of $W\sb b$ is bounded below by $2-\alpha -(C/\ln b)$. We also show that if a function f is convex Lipschitz of order $\alpha$, then the graph of f has $\sigma$- finite measure with respect to Hausdorff's measure in dimension $2- \alpha$. The convex Lipschitz functions of order $\alpha$ include Zygmund's class $\Lambda\sb{\alpha}$. Our analysis shows that the graph of the classical van der Waerden-Takagi nowhere differentiable function has $\sigma$-finite measure with respect to $h(t)=t/\ln (1/t)$.

MSC 2000:: *28A75 Geometric measure theory
42A32 Trigonometric series of special types
26A27 Nondifferentiability of functions of one real variable

Keywords: Hausdorff dimension of the graphs of various continuous functions; fractal dimension; Weierstrass-Mandelbrot functions; convex Lipschitz functions; Zygmund's class; van der Waerden-Takagi nowhere differentiable function

Cited in: Zbl 0803.60079 Zbl 0691.58029 Zbl 0625.58021

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