Nearly k-Distance Sets.

Abstract

We say that a set of points $S\subset R^d$ is an $\epsilon$-nearly k-distance set if there exist $1\le t_1\le \dots \le t_k$, such that the distance between any two distinct points in S falls into $[t_1,t_1+\epsilon ]\cup \cdots \cup [t_k,t_k+\epsilon ]$. In this paper, we study the quantity $\begin{array}{c}{M}_k\left(d\right)=\underset{\epsilon \to 0}{lim}max\left\{\right|S|:S\text{is an}\epsilon \text{-nearly}k\text{-distance set in}{R}^d\}\end{array}$and its relation to the classical quantity $m_k\left(d\right)$: the size of the largest k-distance set in $R^d$. We obtain that $M_k\left(d\right)=m_k\left(d\right)$ 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 $[t_1,t_1+1]\cup \cdots \cup [t_k,t_k+1]$, where $t_1,\cdots ,t_k$ 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.