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Rational Polygons: Odd Compression Ratio and Odd Plane Coverings

  • Rom Pinchasi
  • Yuri Rabinovich
Chapter

Abstract

Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant depending on P alone.

The key ingredient of the proof is a construction of an odd cover of the plane by translates of P. That is, we establish a family \(\mathcal{F}\) of translates of P covering (almost) every point in the plane a uniformly bounded odd number of times.

References

  1. 1.
    R. Pinchasi, Points covered an odd number of times by translates. Am. Math. Mon. 121(7), 632–636 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Oren, I. Pak, R. Pinchasi, On the odd area of planar sets. Discrete Comput. Geom. 55(3), 715–724 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    The International Mathematics Tournament of the Towns, Fall 2009. Available at http://www.math.toronto.edu/oz/turgor/archives/TT2009F_JAproblems.pdf
  4. 4.
    C.A. Rogers, Packings and Coverings. Cambridge Tracts in Mathematics and Mathematical Physics, No. 54 (Cambridge, 1964)Google Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentTechnion–Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Computer ScienceHaifa UniversityHaifaIsrael

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