For any positive integer k and ε > 0, there exist nk,ε, ck, e > 0 with the following property. Given any system of n > nk,ε points in the plane with minimal distance at least 1 and any t1, t2…, tk ≥ 1, the number of those pairs of points whose distance is between ti and [formula omitted] for some 1 ≤ i ≤ k, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.
ASJC Scopus subject areas
- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics