Abstract
AbstractIn this paper we show that if a compact set $$E \subset {\mathbb {R}}^d$$
E
⊂
R
d
, $$d \ge 3$$
d
≥
3
, has Hausdorff dimension greater than $$\frac{(4k-1)}{4k}d+\frac{1}{4}$$
(
4
k
-
1
)
4
k
d
+
1
4
when $$3 \le d<\frac{k(k+3)}{(k-1)}$$
3
≤
d
<
k
(
k
+
3
)
(
k
-
1
)
or $$d- \frac{1}{k-1}$$
d
-
1
k
-
1
when $$\frac{k(k+3)}{(k-1)} \le d$$
k
(
k
+
3
)
(
k
-
1
)
≤
d
, then the set of congruence class of simplices with vertices in E has nonempty interior. By set of congruence class of simplices with vertices in E we mean $$\begin{aligned} \Delta _{k}(E) = \left\{ \textbf{t} = \left( t_{ij} \right) : |x_i-x_j|=t_{ij}; \ x_i,x_j \in E; \ 0\le i < j \le k \right\} \subset {\mathbb {R}}^{\frac{k(k+1)}{2}} \end{aligned}$$
Δ
k
(
E
)
=
t
=
t
ij
:
|
x
i
-
x
j
|
=
t
ij
;
x
i
,
x
j
∈
E
;
0
≤
i
<
j
≤
k
⊂
R
k
(
k
+
1
)
2
where $$2 \le k <d$$
2
≤
k
<
d
. This result improves the previous best results in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of E has nonempty interior when $$d=3$$
d
=
3
as well as extending to all simplices. The present work can be thought of as an extension of the Mattila–Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.
Publisher
Springer Science and Business Media LLC