Nearly k-Distance Sets

We say that a set of points [Formula: see text] is an [Formula: see text] -nearly k-distance set if there exist [Formula: see text] , such that the distance between any two distinct points in S falls into [Formula: see text] . In this paper, we study the quantity [Formula: see text] and its relation to the classical quantity [Formula: see text] : the size of the largest k-distance set in [Formula: see text] . We obtain that [Formula: see text] for [Formula: see text] , as well as for any fixed k, provided that d is sufficiently large. The last result answers a question, proposed by Erdős, Makai, and Pach. We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 90s: given n points in [Formula: see text] , how many pairs of them form a distance that belongs to [Formula: see text] , where [Formula: see text] are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to [Formula: see text] , as well as obtain an exact answer for the same ranges k, d as above.