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Families of plane curves having translates in a set of measure zero

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Eric Sawyer
Eric Sawyer
Affiliation:
Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada. L8S 4K1
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Abstract

We construct a universal function φ on the real line such that, for every continuously differentiable function f the range of f – φ has measure zero. We then apply this to obtain results on curve packing that generalize the Besicovitch set. In particular, we show that given a continuously differentiable family of measurable curves, there exists a plane set of measure zero containing a translate of each curve in the family. Examples are given to show that the differentiability hypothesis cannot be weakened to a Lipschitz condition of order α for any 0<α<1.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 34 , Issue 1 , June 1987 , pp. 69 - 76
Copyright
Copyright © University College London 1987

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