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Transactions of the American Mathematical Society

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On the Hausdorff dimension of some graphs


Authors: R. Daniel Mauldin and S. C. Williams
Journal: Trans. Amer. Math. Soc. 298 (1986), 793-803
MSC: Primary 28A75; Secondary 42A32
DOI: https://doi.org/10.1090/S0002-9947-1986-0860394-7
MathSciNet review: 860394
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Abstract: Consider the functions

$\displaystyle {W_b}(x) = \sum\limits_{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],} $

where $ b > 1$, $ 0 < \alpha < 1$, each $ {\theta _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_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 _\alpha }$. Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has $ \sigma $-finite measure with respect to $ h(t) = t/\ln (1/t)$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0860394-7
Keywords: Fractal dimension, Hausdorff dimension, Weierstrass-Mandelbrot functions
Article copyright: © Copyright 1986 American Mathematical Society

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