Distance sets of well-distributed planar point sets (Q1880213)

For \(K\) a balanced convex compact set in \(\mathbb{R}^2\), the set \(S\subset \mathbb{R}^2\) is \(K\)-well distributed if any point of \(\mathbb{R}^2\) lies within a uniformly bounded \(K\)-distance from \(S\). The paper studies for such \(S\) the asymptotic behavior of \(\Delta_N(S)\), the number of distinct \(K\)-distances \(\leq N\) between pairs in \(S\). E.g. it is shown that if \(\Delta_N(S)=O(N^{1.5})\) then \(K\) is polygonal.