Nearly k-Distance Sets

We say that a set of points $$ is an $$-nearly k-distance set if there exist $$, such that the distance between any two distinct points in S falls into $$. In this paper, we study the quantity and its relation to the classical quantity $$: the size of the largest k-distance set in $$. We obtain that $$ for $$, 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 $$, how many pairs of them form a distance that belongs to $$, where $$ are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to $$, as well as obtain an exact answer for the same ranges k, d as above.