## On the Hausdorff dimension of some graphs

HTML articles powered by AMS MathViewer

- by R. Daniel Mauldin and S. C. Williams
- Trans. Amer. Math. Soc.
**298**(1986), 793-803 - DOI: https://doi.org/10.1090/S0002-9947-1986-0860394-7
- PDF | Request permission

## Abstract:

Consider the functions \[ {W_b}(x) = \sum \limits _{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],} \] 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 $h(t) = t/\ln (1/t)$.## References

- M. V. Berry and Z. V. Lewis,
*On the Weierstrass-Mandelbrot fractal function*, Proc. Roy. Soc. London Ser. A**370**(1980), no. 1743, 459–484. MR**570931**, DOI 10.1098/rspa.1980.0044
A. S. Besicovitch and H. D. Ursell, - A. M. Bruckner and K. M. Garg,
*The level structure of a residual set of continuous functions*, Trans. Amer. Math. Soc.**232**(1977), 307–321. MR**476939**, DOI 10.1090/S0002-9947-1977-0476939-X - K. J. Falconer,
*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284** - G. H. Hardy,
*Weierstrass’s non-differentiable function*, Trans. Amer. Math. Soc.**17**(1916), no. 3, 301–325. MR**1501044**, DOI 10.1090/S0002-9947-1916-1501044-1
F. Hausdorff, - James L. Kaplan, John Mallet-Paret, and James A. Yorke,
*The Lyapunov dimension of a nowhere differentiable attracting torus*, Ergodic Theory Dynam. Systems**4**(1984), no. 2, 261–281. MR**766105**, DOI 10.1017/S0143385700002431 - Benoit B. Mandelbrot,
*Fractals: form, chance, and dimension*, Revised edition, W. H. Freeman and Co., San Francisco, Calif., 1977. Translated from the French. MR**0471493** - J. Moser,
*On a theorem of Anosov*, J. Differential Equations**5**(1969), 411–440. MR**238357**, DOI 10.1016/0022-0396(69)90083-7 - C. A. Rogers,
*Hausdorff measures*, Cambridge University Press, London-New York, 1970. MR**0281862**
A. Zygmund,

*Sets of fractional dimensions*, V:

*On dimensional numbers of some continuous curves*, J. London Math. Soc. (2)

**32**(1937), 142-153.

*Dimension and ausseres Mass*, Math. Ann.

**79**(1919), 157-179.

*Trigonometric series*, Cambridge Univ. Press, Cambridge, 1968.

## Bibliographic Information

- © Copyright 1986 American Mathematical Society
- 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