Families of plane curves having translates in a set of measure zero Academic Article uri icon

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abstract

  • AbstractWe 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.

publication date

  • June 1987