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Combinatorica

, Volume 16, Issue 3, pp 321–324 | Cite as

A remark on sets having the Steinhaus property

  • Mihai Ciucu
Article

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.

Mathematics Subject Classification (1991)

05 B 40 52 C 15 11 H 16 

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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.Google Scholar
  2. [2]
    S. Lang: Real and Functional Analysis, Graduate Texts in Mathematics, Vol. 142, Springer Verlag, Berlin, Heidelberg, New York, 1993.Google Scholar
  3. [3]
    W. Sierpinski: Sur une probleme de H. Steinhaus concernant les ensembles de points sur le plan,Fund. Math. 46 (1959), 191–194.Google Scholar

Copyright information

© Akadémiai Kiadó 1996

Authors and Affiliations

  • Mihai Ciucu
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor

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