On multiplicities of interpoint distance

Given a set X subset of R$^2$ of points and a distance d > 0, the multiplicity of is the number of times the distance appears between points in X. Let a(1)(X) >= a(2)(X) >= >= a(m)(X) denote the multiplicities of the distances determined by X and let a(X) = (a(1)(X), . . . , a(m)(X)). In this paper, we study several questions from Erd & odblac;s's time regarding distance multiplicities. Among other results, we show that: (1) If is convex or "not too convex", then there exists a distance other than the diameter that has multiplicity at most n. (2) There exists a set X subset of R-2 of points, such that many distances occur with high multiplicity. In particular, at least n(ohm(1/log log n)) distances have superlinear multiplicity in n. (3) For any (not necessarily fixed) integer 1 <= k <= log n, there exists X subset of R-2 of points, such that the difference between the k(th) and (k+ 1)(th) and largest multiplicities is at least ohm(n log n/k). Moreover, the distances in X with the largest k multiplicities can be prescribed. (4) For every n is an element of N, there exists X subset of R-2 of n points, not all collinear or cocircular, such that a(X) = (n-1, n-2, . ... mehr. . ,1).There also exists Y subset of R-2 of points with pairwise distinct distance multiplicities and a(Y)not equal (n-1, n-2, . . . ,1)