TITLE:
Non-Spectrality of Certain Self-Affine Measures on the Generalized Spatial Sierpinski Gasket
AUTHORS:
Yongli Hu, Zhicheng Zhang, Qi Wang
KEYWORDS:
Sierpinski Gasket, Non-Spectrality, Orthogonal Exponentials, Digit Set
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.11,
November
29,
2024
ABSTRACT: Let
μ
M,D
be a self-affine measure associated with an expanding integer matrix
M=[
p
1
,0,0;
p
4
,
p
2
,0;
p
5
,0,
p
3
]
and the digit set
D={
0,
e
1
,
e
2
,
e
3
}
in the space
R
3
, where
p
1
,
p
2
,
p
3
∈Z\{
0,±1 }
,
p
4
,
p
5
∈Z
and
e
1
,
e
2
,
e
3
are the standard basis of unit column vectors in
R
3
. In this paper, we mainly consider the case
p
1
,
p
2
,
p
3
∈2Z+1,
p
2
≠
p
3
,
p
4
=l(
p
1
−
p
2
),
p
5
=l(
p
3
−
p
1
),
where
l∈2Z
. We prove that
μ
M,D
is a non-spectral measure, and there are at most 4-element
μ
M,D
-orthogonal exponentials, and the number 4 is the best. The results here generalize the known results.