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The packing measure of the graphs and level sets of certain continuous functions

Published online by Cambridge University Press:  24 October 2008

Fraydoun Rezakhanlou
Fraydoun Rezakhanlou
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A.
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Abstract

The relationship between the local growth of a continuous function and the packing measure of its level sets and of its graph is studied. For the Weierstrass function with b an integer such that b ≥ 2 and with 0 < α < 1, and for x ∈ Range (W) outside a set of first category, the level set W−1(x) has packing dimension at least 1 − α. Furthermore, for almost all x ∈ Range (W), the packing dimension of f is at most 1 − α. Finer results on the occupation measure and the size of the graph of a continuous function satisfying the Zygmund Λ-condition are obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 347 - 360
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Anderson, J. M. and Pitt, L. D.. Probabilistic Behavior of Functions in the Zygmund Spaces Λ* and λ*. (Preprint.)Google Scholar
[2]Berry, M. V. and Lewis, Z. V.. On the Weierstrass–Mandelbrot fractal function. Proc. Roy. Soc. London Ser. A 370 (1980), 459–484.Google Scholar
[3]Besicovitch, A. S. and Ursell, H. D.. Sets of fractional dimensions, V: On dimensional numbers of some continuous curves. J. London Math. Soc. 12 (1937), 18–25.CrossRefGoogle Scholar
[4]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[5]Izumi, M., Izumi, S.-I. and Kahane, J.-P.. Théorèmes elementaires sur les séries de Fourier lacunaires. J. Analyse Math. 14 (1965), 235–246.CrossRefGoogle Scholar
[6]Kahane, J.-P.. Geza Freud and lacunary Fourier series. J. Approx. Theory 46 (1986), 51–57.CrossRefGoogle Scholar
[7]Marstrand, J. M.. The dimension of cartesian product sets. Proc. Cambridge Philos. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[8]Mauldin, R. D. and Williams, S. C.. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793–803.CrossRefGoogle Scholar
[9]Orey, S. and Taylor, S. J.. How often on a Brownian path does the law of iterated logarithm fail? Proc. London Math. Soc. 28 (1974), 174–192.CrossRefGoogle Scholar
[10]Rezakhanlou, F. and Taylor, S. J.. The packing measure of the graph of a stable process. (Preprint.)Google Scholar
[11]Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
[12]Taylor, S. J.. The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986), 383–406.CrossRefGoogle Scholar
[13]Taylor, S. J.. The use of packing measure in the analysis of random sets. In Proceedings of the 15th Symposium on Stochastic Processes and Applications, Lecture Notes in Math. vol. 1203 (Springer-Verlag, 1986), pp. 214–222.CrossRefGoogle Scholar
[14]Taylor, S. J. and Tricot, C.. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[15]Zygmund, A.. Trigonometric Series, 2nd edn. (Cambridge University Press, 1959).Google Scholar