Variations on the theme of repeated distances (Q756139)

Given a set \(X=\{x_1,...x_n\}\) of n points in \({\mathbb{R}}^d\), let \(f(X)\) denote the number of pairs \(\{x_i,x_j\}\) whose Euclidean distances \(||x_i-x_j|| = 1\). Let \[ f_d(n) = \max_{X\subset {\mathbb{R}}^d,\quad |X| \leq n}f(X). \] An asymptotically sharp estimate for the error term in this maximum is given for \(d\geq 4\). Tight upper bounds are also determined for the total number of occurrences of what are called ``favourite'' distances from \(n\) points in \({\mathbb{R}}^d\), \(d\geq 4\). Some related results are also proved for distances determined by \(n\) disjoint compact convex sets in \({\mathbb{R}}^2\).