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Tangents of σ-finite curves and scaled oscillation

Published online by Cambridge University Press:  28 January 2016

MARIANNA CSÖRNYEI  and
BOBBY WILSON
MARIANNA CSÖRNYEI
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, U.S.A. e-mail: csornyei@math.uchicago.edu; bobbylew@math.uchicago.edu
BOBBY WILSON
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, U.S.A. e-mail: csornyei@math.uchicago.edu; bobbylew@math.uchicago.edu
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Abstract

We show that every continuous simple curve with σ-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 1 - 12
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1]Alberti, G., Csörnyei, M. and Preiss, D.Paper in preparation.Google Scholar
[2]Balogh, Z. M. and Csörnyei, M.Scaled-oscillation and regularity. Proc. Arer Math. Soc. 134, No. 9 (2006), 2667–2675.Google Scholar
[3]Besicovitch, A. S.On the definition of tangents to sets of infinite linear measure. Proc. Camb. Phil. Soc. 52 (1956), 20–29.Google Scholar
[4]Besicovitch, A. S.Analysis of tangential properties of curves of infinite length. Proc. Camb. Phil. Soc. 53 (1957), 69–72.Google Scholar
[5]Falconer, K. J.Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[6]Hanson, B.Linear dilatation and differentiability of homeomorphisms of ${\mathbb R}^{n}$. Proc. Amer. Math. Soc. 140 (2012), 3541–3547.Google Scholar
[7]Mattila, P.Geometry of sets and measures in Euclidean spaces (Cambridge University Press, 1995).Google Scholar
[8]Roger, F.Sur la relation entre les propriétés tangentielles et métriques des ensembles cartésiens. C. R. Math. Acad. Sci. Paris. 201 (1935), 871–873.Google Scholar
[9]Saks, S.Theory of the Integral (Dover, 1964).Google Scholar
[10]Stein, E.Singular Integrals and Differentiability Properties of Functions (Princeton University Press, 1970).Google Scholar