@article{9651,
  abstract     = {We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞). Two separated nets are called ϕ-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R). We show that the spectrum of ϕ-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation, coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞. We further undertake a comparison of our notion of ϕ-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz equivalence.},
  author       = {Dymond, Michael and Kaluza, Vojtech},
  issn         = {1572-9168},
  journal      = {Geometriae Dedicata},
  publisher    = {Springer Nature},
  title        = {{Divergence of separated nets with respect to displacement equivalence}},
  doi          = {10.1007/s10711-023-00862-3},
  year         = {2023},
}

@article{4080,
  abstract     = {This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least n12−112+O(1)n47  points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k.},
  author       = {Edelsbrunner, Herbert and Hasan, Nany and Seidel, Raimund and Shen, Xiao},
  issn         = {1572-9168},
  journal      = {Geometriae Dedicata},
  number       = {1},
  pages        = {1 -- 12},
  publisher    = {Springer},
  title        = {{Circles through two points that always enclose many points}},
  doi          = {10.1007/BF00181432},
  volume       = {32},
  year         = {1989},
}