On distinct distances and λ-free point sets

It is generally believed that the minimum number of distinct distances determined by a set of n points in the Euclidean space is attained by sets having a very regular grid-like structure: for instance n equidistant points on the line, or a root n x root n section of the integer grid in the plane. What happens if we perturb the regularity of the grid, say by not allowing two points together with their midpoint to be in the set? Do we get more distances in a set of n points? In particular, is this number linear for such a set of n points in the plane? We call a set of points midpoint-free if no point is the midpoint of two others. More generally, let lambda epsilon (0, 1) be a fixed rational number. We say that a set of points P is lambda-free if for any triple of distinct points a, b, c epsilon P, we have lambda a + (1 - lambda)b not equal c. We first make a investigation of midpoint-free (more generally lambda-free) sets on the line with respect to the number of distinct distances determined by a set of n points and provide such estimates. Other related distance problems are also discussed, including possible implications of obtaining good estimates on the minimum number of distinct distances in a lambda-free point set in the plane for the general problem of distinct distances in the plane. (C) 2007 Elsevier B.V. All rights reserved.