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.