How likely can a point be in different Cantor sets

• Published in: Nonlinearity Vol. 35; no. 3; pp. 1402 - 1430
• Main Authors: Jiang, Kan, Kong, Derong, Li, Wenxia
• Format: Journal Article
• Language: English
• Published: IOP Publishing 03.03.2022
• Subjects: Hausdorff dimension; intersection of Cantor sets; self-similar set; thickness of a Cantor set
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/ac4b3c

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.