(1) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;(2) Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Abstract:
We prove that a well-distributed subset of ${Bbb R}^2$can have a distance set $Delta$ with $#(Deltacap [0,N])leqCN^{3/2-epsilon}$ 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.
