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Density and covering properties of intervals of ℝ n
Author(s) -
Keleti Tamás
Publication year - 2000
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015849
Subject(s) - mathematics , combinatorics , unit cube , rectangle , intersection (aeronautics) , lebesgue integration , unit interval , lebesgue measure , unit square , interval (graph theory) , sequence (biology) , class (philosophy) , unit (ring theory) , measure (data warehouse) , discrete mathematics , geometry , mathematics education , database , artificial intelligence , biology , computer science , engineering , genetics , aerospace engineering
The key result of this paper proves the existence of functions ρ n ( h ) for which, whenever H is a (Lebesgue) measurable subset of the n ‐dimensional unit cube I n with measure | H | > h and ℛ is a class of subintervals ( n ‐dimensional axis‐parallel rectangles) of I n that covers H , then there exists an interval R ∈ℛ in which the density of H is greater than ρ n ( h ); that is, | H ∩ R |/| R |>ρ n ( h ) (=( h/2n ) 2 ). It is shown how to use this result to find 4 points of a measurable subset of the unit square which are the vertices of an axis‐parallel rectangle that has quite large intersection with the original set. Density and covering properties of classes of subsets of ℝ n are introduced and investigated. As a consequence, a covering property of the class of intervals of ℝ n is obtained: if ℛ is a family of n ‐dimensional intervals with| ∪ R | < ∞ , then there is a finite sequence R 1 , …, R m ∈ℛ such that|∪ k = 1 mR k| ⩾ ( 1 − ε ) | ∪ R |and‖∑ k = 1 mχ R k‖ q ⩽ C ( n , q , ε )| ∪ R |1 / q.