Author:
Jiang Kan,Kong Derong,Li Wenxia
Abstract
Abstract
Given
m
∈
N
⩾
2
, let
K
=
K
λ
:
λ
∈
(
0
,
1
/
m
]
be a class of self-similar sets with each
K
λ
=
∑
i
=
1
∞
d
i
λ
i
:
d
i
∈
{
0
,
1
,
…
,
m
−
1
}
,
i
⩾
1
. In this paper we investigate the likelyhood of a point in the self-similar sets of
K
. More precisely, for a given point x ∈ (0, 1) we consider the parameter set
Λ
(
x
)
=
λ
∈
(
0
,
1
/
m
]
:
x
∈
K
λ
, and show that Λ(x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets of Λ(x) with large thickness we show that for any x, y ∈ (0, 1) the intersection Λ(x) ∩ Λ(y) also has full Hausdorff dimension.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
2 articles.
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1. On a class of self-similar sets which contain finitely many common points;Proceedings of the Royal Society of Edinburgh: Section A Mathematics;2024-05-30
2. How inhomogeneous Cantor sets can pass a point;Mathematische Zeitschrift;2022-08-23