Upper and lower densities have the strong Darboux property

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• 作者：Paolo Leonetti^a ; ^{leonetti.paolo@gmail.com} ; Salvatore Tringali^b ; ¹ ; ^{salvatore.tringali@uni-graz.at}
• 关键词：primary ; 11B05 ; 28A10 ; secondary ; 39B62 ; 60B99
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：445-455
• 全文大小：344 K
• 卷排序：174

文摘

Let P(N)P(N) be the power set of N. An upper density (on N) is a nondecreasing and subadditive function μ⋆:P(N)→Rμ⋆:P(N)→R such that μ⋆(N)=1μ⋆(N)=1 and μ⋆(k⋅X+h)=1kμ⋆(X) for all X⊆NX⊆N and h,k∈N+h,k∈N+, where k⋅X+h:={kx+h:x∈X}k⋅X+h:={kx+h:x∈X}.The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper Pólya, and upper analytic densities are some examples of upper densities.We show that every upper density μ⋆μ⋆ has the strong Darboux property, and so does the associated lower density, where a function f:P(N)→Rf:P(N)→R is said to have the strong Darboux property if, whenever X⊆Y⊆NX⊆Y⊆N and a∈[f(X),f(Y)]a∈[f(X),f(Y)], there is a set A   such that X⊆A⊆YX⊆A⊆Y and f(A)=af(A)=a. In fact, we prove the above under the assumption that the monotonicity of μ⋆μ⋆ is relaxed to the weaker condition that μ⋆(X)≤1μ⋆(X)≤1 for every X⊆NX⊆N.