Nearly k-distance sets

Abstract

We say that $S\subset \mathbb{R}^d$ is an \emph{$\varepsilon$-nearly $k$-distance set} if there exist $1\le t_1\le \ldots\le t_k$ such that the distance between any two distinct points of $S$ falls into $[t_1,t_1+\varepsilon]\cup\ldots\cup[t_k,t_k+\varepsilon]$. In this abstract, we propose to study the quantity $M_k(d) := \lim_{\varepsilon\to 0}\max\{|S|\ :\ S\text{ is an }\varepsilon\text{-nearly $k$-distance set in }\mathbb{R}^d\}$. Let $m_k(d)$ be the maximal cardinality of a $k$-distance set in $\mathbb{R}^d$. We show that $M_k(d) = m_k(d)$ if either $d\ge d(k)$ or $k\le 3$. We also address a closely related Tur\'an-type problem, studied by Erd\H os, Makai, Pach, and Spencer in the 80's: given $n$ points in $\mathbb{R}^d$, how many pairs out of them form a distance that belongs to $[t_1,t_1+1]\cup\ldots\cup[t_k,t_k+1],$ where $t_1,\ldots, t_k$ are fixed and any two points in the set are at distance at least $1$ apart? We obtain an exact answer for the same $k,d$ as above.