Nearly Equal Distances in the Plane

For any positive integer k and ε > 0, there exist n^k,ε, c^{k, e} > 0 with the following property. Given any system of n > n_k,ε points in the plane with minimal distance at least 1 and any t₁, t₂…, t_k ≥ 1, the number of those pairs of points whose distance is between t_i and [formula omitted] for some 1 ≤ i ≤ k, is at most (n²/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.