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Sets with Large Intersection Properties
Author(s) -
Falconer K. J.
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/49.2.267
Subject(s) - mathematics , hausdorff dimension , countable set , intersection (aeronautics) , combinatorics , class (philosophy) , dimension (graph theory) , packing dimension , urysohn and completely hausdorff spaces , discrete mathematics , hausdorff measure , minkowski–bouligand dimension , fractal dimension , fractal , computer science , mathematical analysis , artificial intelligence , engineering , aerospace engineering
For 0 < s ⩽ n let s be the class of G δ ‐subsets of R n such that F ∈ s if∩ i − 1 ∞f i ( F ) has Hausdorff dimension at least s for all sequences of similarity transformations{ f 1 } i − 1 ∞ . We show that s is closed under countable intersections and under bi‐Lipschitz functions, and thus is the maximal class of G δ ‐sets of Hausdorff dimension at least s that is closed under countable intersection and similarities. We also show that sets in s must have packing dimension n . Many examples of s ‐sets occur in Diophantine approximation.