Nearly k -Distance Sets.

We say that a set of points S ⊂ R d is an ε -nearly k -distance set if there exist 1 ≤ t 1 ≤ … ≤ t k , such that the distance between any two distinct points in S falls into [ t 1 , t 1 + ε ] ∪ ⋯ ∪ [ t k , t k + ε ] . In this paper, we study the quantity M k ( d ) = lim ε → 0 max { | S | : S is an ε -nearly k -distance set in R d } and its relation to the classical quantity m k ( d ) : the size of the largest k -distance set in  R d . We obtain that M k ( d ) = m k ( d ) for k = 2 , 3 , 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  R d , how many pairs of them form a distance that belongs to [ t 1 , t 1 + 1 ] ∪ ⋯ ∪ [ t k , t k + 1 ] , where t 1 , ⋯ , t k 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 M k ( d - 1 ) , as well as obtain an exact answer for the same ranges k ,  d as above.