Popular distances in 3-space Original Research Article

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn1/3 log log n ⩽ m(n) ⩽ n3/5 β(n), where c > 0 is a constant and β(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.