Intersections of middle-α Cantor sets with a fixed translation. (1st March 2023)

• Record Type: Journal Article
• Authors: Huang, Yan; Kong, Derong
• Is Part Of: Nonlinearity. Volume 36:Number 3(2023)
• Page Start: 1461
• Page End: 1490
• Publication Date: 2023-03-01
• Subjects: intersection -- Cantor set -- Hausdorff dimension -- packing dimension -- level set -- digit frequency; 28A78 -- 37A10 -- Secondary: 28D10 -- 28A80 -- 37B10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/acb39a
• Languages: English
• ISSNs: 0951-7715

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.