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Comparing Packing Measures to Hausdorff Measures on the Line
Author(s) -
Feng DeJun
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200207)241:1<65::aid-mana65>3.0.co;2-i
Subject(s) - mathematics , hausdorff measure , infimum and supremum , hausdorff space , outer measure , measure (data warehouse) , hausdorff distance , line (geometry) , combinatorics , discrete mathematics , hausdorff dimension , mathematical analysis , geometry , data mining , fractal , computer science , minkowski–bouligand dimension , fractal dimension
For each 0 < s < 1, define$$c(s) = \inf _{E} {{{\cal P}^s (E)} \over {{\cal H} ^s (E)}} \, ,$$where ${\cal P}^s$ , ${\cal H}^s$ denote respectively the s ‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with $0 < {\cal H}^s (E) < \infty$ . In this paper we give a nontrivial estimation of c ( s ), namely, $2^s (1+v(s))^s \le c(s) \le 2^s \left (2 ^{1 \over s} - 1 \right )$ for each 0 < s < 1, where $v(s) = \min \left \{ 16 ^{- {{1} \over {1-s}}}, 8 ^{- {{1} \over {(1-s)^2}}} \right \}$ . As an application, we obtain a lower density theorem for Hausdorff measures.