Nearly $k$-distance sets

We say that a set of points $S\subset \mathbb{R}^d$ is an $\varepsilon$-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 in $S$ falls into $[t_1,t_1+\varepsilon]\cup\ldots\cup[t_k,t_k+\varepsilon]$. In this paper, we study the quantity $M_k(d) = \lim_{\varepsilon\to 0}\max\{|S|\ :\ S\text{ is an }\varepsilon\text{-nearly } k \text{-distance set in } \mathbb{R}^d\}$ and its relation to the classical quantity $m_k(d)$: the size of the largest $k$-distance set in $\mathbb{R}^d$. 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 Erd\H{o}s, Makai and Pach. We also address a closely related Tur\'an-type problem, studied by Erd\H{o}s, Makai, Pach, and Spencer in the 80's: given $n$ points in $\mathbb{R}^d$, how many pairs 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 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.