Popular distances in 3-space (Q1301635)

The main problem of this paper is: What is the smallest integer \(m(n)\) with the property that any set of \(n\) points in Euclidean 3-space has an element such that at most \(m\) other points are equidistant from it? The authors obtain the following valuation: \[ c n^{1/3} \log\log n\leq m(n) \leq n^{3/5} \beta(n), \] where \(c>0\) is a constant and \(\beta(n)\) is an extremely slowly growing function, related to the inverse of the Ackermann function.