Abstract
We prove that a well-distributed subset of ${\Bbb R}^2$ can have a distance set $\Delta$ with $\#(\Delta\cap [0,N])\leq CN^{3/2-\epsilon}$ only if the distance is induced by a polygon $K$. Furthermore, if the above estimate holds with $\epsilon=\frac12$, then $K$ can have only finitely many sides.
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Iosevich, A., Laba, I. Distance Sets of Well-Distributed Planar Point Sets. Discrete Comput Geom 31, 243–250 (2004). https://doi.org/10.1007/s00454-003-2857-1
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DOI: https://doi.org/10.1007/s00454-003-2857-1