Abstract
Abstract
For
λ
∈
(
0
,
1
/
3
]
let C
λ
be the middle-
(
1
−
2
λ
)
Cantor set in
R
. Given
t
∈
[
−
1
,
1
]
, excluding the trivial case we show that
Λ
(
t
)
:=
λ
∈
(
0
,
1
/
3
]
:
C
λ
∩
(
C
λ
+
t
)
≠
∅
is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of
Λ
(
t
)
, which reveals a dimensional variation principle. Furthermore, for any
β
∈
[
0
,
1
]
we show that the level set
Λ
β
(
t
)
:
=
{
λ
∈
Λ
(
t
)
:
dim
H
(
C
λ
∩
(
C
λ
+
t
)
)
=
dim
P
(
C
λ
∩
(
C
λ
+
t
)
)
=
β
log
2
−
log
λ
}
has equal Hausdorff and packing dimension
(
−
β
log
β
−
(
1
−
β
)
log
1
−
β
2
)
/
log
3
. We also show that the set of
λ
∈
Λ
(
t
)
for which
dim
H
(
C
λ
∩
(
C
λ
+
t
)
)
≠
dim
P
(
C
λ
∩
(
C
λ
+
t
)
)
has full Hausdorff dimension.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics