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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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References

1.Besicovitch, A. S.. Sur deux questions d'integrabilite des fonctions. J. Soc. Phys.-Math., Perm, 2 (1919), 105–123.Google Scholar
2.Besicovitch, A. S.. On the fundamental geometric properties of linearly measurable plane sets of points. Math. Annalen, 98 (1928), 422–464.Google Scholar
3.Besicovitch, A. S.. On Kakeya's problem and a similar one. Math. Zeit., 27 (1928), 312–320.CrossRefGoogle Scholar
4.Besicovitch, A. S.. On fundamental geometric properties of line sets. J. London Math. Soc, 39 (1964), 441–448.CrossRefGoogle Scholar
5.Besicovitch, A. S. and Rado, R.. A plane set of measure zero containing circumferences of every radius. J. London Math. Soc., 43 (1968), 717–719.CrossRefGoogle Scholar
6.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
7.Davies, R. O.. Another thin set of circles. J. London Math. Soc, 5 (1972), 191–192.Google Scholar
8.Falconer, K. J.. Sections of sets of zero Lebesgue measure. Mathematika, 27 (1980), 90–96.CrossRefGoogle Scholar
9.Falconer, K. J.. Hausdorff dimension and the exceptional set of projections. Mathematika, 29 (1982), 109–115.CrossRefGoogle Scholar
10.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
11.Kahane, J.-P.. Trois notes sur les ensembles parfaits lineaires. I'Enseignement Math., 15 (1969), 185–192.Google Scholar
12.Kinney, J. R.. A thin set of circles. Amer. Math. Monthly, 75 (1968), 1077–1081.CrossRefGoogle Scholar
13.Kline, S. A.. On curves of fractional dimensions. J. London Math. Soc., 20 (1945), 79–86.Google Scholar
14.Marstrand, J. M.. An application of topological transformation groups to the Kakeya problem. Bull. London Math. Soc., 4 (1972), 191–195.CrossRefGoogle Scholar
15.Marstrand, J. M.. Packing smooth curves in Rq. Mathematika, 26 (1979), 1–12.Google Scholar
16.Marstrand, J. M.. Packing planes in R3. Mathematika, 26 (1979), 1–12.CrossRefGoogle Scholar
17.Rogers, C. A.. Hausdorff Measures (University Press, Cambridge, 1970).Google Scholar