Nearly k-Distance Sets

Abstract

We say that a set of points $$S\subset {{\mathbb {R}}}^d$$ S ⊂ R d is an $$\varepsilon $$ ε -nearly k-distance set if there exist $$1\le t_1\le \ldots \le t_k$$ 1 ≤ t 1 ≤ … ≤ t k , such that the distance between any two distinct points in S falls into $$[t_1,t_1+\varepsilon ]\cup \cdots \cup [t_k,t_k+\varepsilon ]$$ [ t 1 , t 1 + ε ] ∪ ⋯ ∪ [ t k , t k + ε ] . In this paper, we study the quantity $$\begin{aligned} M_k(d) = \lim _{\varepsilon \rightarrow 0}\max {\{|S|:S\,\text { is an}\, \varepsilon \text {-nearly}\, k\text {-distance set in}\,{{\mathbb {R}}}^d\}} \end{aligned}$$ 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 $$m_k(d)$$ m k ( d ) : the size of the largest k-distance set in $${{\mathbb {R}}}^d$$ R d . We obtain that $$M_k(d)=m_k(d)$$ M k ( d ) = m k ( d ) for $$k=2,3$$ 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 $${{\mathbb {R}}}^d$$ R d , how many pairs of them form a distance that belongs to $$[t_1,t_1+1]\cup \cdots \cup [t_k,t_k+1]$$ [ t 1 , t 1 + 1 ] ∪ ⋯ ∪ [ t k , t k + 1 ] , where $$t_1,\dots ,t_k$$ 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 $$M_k(d-1)$$ M k ( d - 1 ) , as well as obtain an exact answer for the same ranges k, d as above.