UNIFORMLY DISTRIBUTED DISTANCES: A GEOMETRIC APPLICATION OF JANSON’S INEQUALITY

JÁNOS PACH JOEL SPENCER

Abstract. Let d_{1} ≤ d_{2} ≤ … ≤ d\binom{n}{2} denote the distances determined by n points

in the plane. It is shown that min ∑_{i}(d_{i+1}−d_{i})^{2} = O(n^{−6/7}), where the minimum is taken over all point sets with minimal distance d_{1} ≥ 1. This bound is asymptotically tight.