  A set $\cal P$ of $n$ points in $R^d$ is separated if all distances of
distinct points are at least~$1$. Then we may ask how many of these distances,
with multiplicity, lie in an interval $[t, t + 1]$. The authors and J. Spencer
proved that the maximum is $(n^2/2)(1 - 1/d) + O(1)$. The authors showed that
for $d = 2$ and $\cal P$ separated, the maximal number of distances, with
multiplicity, in the union of $k$ unit intervals is $(n^2/2)$ $(1 - 1/(k + 1) +
o(1))$. (In these papers the unit intervals could be replaced by intervals of
length $\text{const}_d\cdot n^{1/d}$.) In this paper we show that for $k = 2$,
and for any $n$, this maximal number is $(n^2/2)(1 - 1/m_{d - 1} + o(1))$,
where $m_{d - 1}$ is the maximal size of a two-distance set in $R^{d - 1}$.
(The value of $m_{d - 1}$ is known for $d - 1 \leq 8$, and for each $d$ it lies
in $\left[\left({d\atop 2}\right), \left({d + 1\atop 2}\right)\right]$. For $d
\neq 4,5$ we can replace unit intervals by intervals of length $\text{const}_d
\cdot n^{1/d}$, and the maximum is the respective Tur\'an number, for $n \geq
n(d)$.) We also investigate a variant of this question, namely with $k$
intervals of the form $[t, t(1 + \varepsilon)]$, for $\varepsilon < \varepsilon
(d, k)$, and for $n > n(d, k)$. Here the maximal number of distances, with
multiplicity, in the union of $k$ such intervals is the Tur\'an number $T(n, (d
+ 1)^k + 1)$. Several of these results were announced earlier by
Makai-Pach-Spencer.
