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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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