A remark on sets having the Steinhaus property

Abstract

A point set satisfies the Steinhaus property if no matter how it is placed on a plane, it covers exactly one integer lattice point. Whether or not such a set exists, is an open problem. Beck has proved [1] that any bounded set satisfying the Steinhaus property is not Lebesgue measurable. We show that any such set (bounded or not) must have empty interior. As a corollary, we deduce that closed sets do not have the Steinhaus property, fact noted by Sierpinski [3] under the additional assumption of boundedness.

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References

  1. [1]

    J. Beck: On a lattice-point problem of H. Steinhaus,Studia Sci. Math. Hungar.,24 (1989), 263–268.

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  2. [2]

    S. Lang: Real and Functional Analysis, Graduate Texts in Mathematics, Vol. 142, Springer Verlag, Berlin, Heidelberg, New York, 1993.

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  3. [3]

    W. Sierpinski: Sur une probleme de H. Steinhaus concernant les ensembles de points sur le plan,Fund. Math. 46 (1959), 191–194.

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Ciucu, M. A remark on sets having the Steinhaus property. Combinatorica 16, 321–324 (1996). https://doi.org/10.1007/BF01261317

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Mathematics Subject Classification (1991)

  • 05 B 40
  • 52 C 15
  • 11 H 16