Abstract
The main result in the paper is a construction of a simple (in fact, just a union of two squares) set T in the plane with the following property. For every \(\varepsilon >0\) there is a family \(\mathcal{F}\) of an odd number of translates of T such that the area of those points in the plane that belong to an odd number of sets in \(\mathcal{F}\) is smaller than \(\varepsilon \).
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References
Pak, I.: Lectures on Discrete Geometry and Convex Polyhedra. Cambridge University Press, Cambridge (to appear). http://www.math.ucla.edu/~pak/geompol8
Pinchasi, R.: Points covered an odd number of times by translates. Am. Math. Mon. 121(7), 632–636 (2014)
Pinchasi, R., Rabinovich, U.: Rational Polygons: Odd Area and Odd Plane Coverings (preprint)
The International Mathematics Tournament of the Towns, Fall of 2009. http://www.math.toronto.edu/oz/turgor/archives/TT2009F_JAproblems
Acknowledgments
Rom Pinchasi was Supported by ISF Grant (Grant No. 1357/12).
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Oren, A., Pak, I. & Pinchasi, R. On the Odd Area of Planar Sets. Discrete Comput Geom 55, 715–724 (2016). https://doi.org/10.1007/s00454-016-9768-4
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DOI: https://doi.org/10.1007/s00454-016-9768-4