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Asymptotic densities of sets of positive integers. (English) Zbl 0516.10052

A density measure is any finitely additive measure which extends the asymptotic density \(d\) to all subsets of \(\mathbb N\). Let \(A(n) = \sum_{a\in A,\ a\le n} 1\). A set function \(\nu\) is said to have property P if for every pair \(A,B \subset \mathbb N\) with \(\lim A(n)/B(n) = t\) then \(\nu(A)/\nu(B) =t\). The authors show that any density measure on \(2^{\mathbb N}\) has property P. By identifying elements of \(2^{\mathbb N}\) with points in \([0,1]\) they show that if \(\nu\) is any set function on \(2^{\mathbb N}\) with property P then if \(A_x\subset A = A_1\), \(\nu(A_x) = \nu(A)/2\) for almost all \(x\in [0,1]\).
Other measure theoretic and topological properties associated with this identification are also discussed.

MSC:

11B05 Density, gaps, topology
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11B83 Special sequences and polynomials
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References:

[1] BANACH S.: Théorie des opérations linéaires. Warszawa 1932. · Zbl 0005.20901
[2] BUCK R. C.: The measure theoretic approach to density. Amer. J. Math. 68, 1946, 560-580. · Zbl 0061.07503
[3] BUCK R. C: Generalized asymptotic density. Amer. J. Math. 75, 1953, 335-346. · Zbl 0050.05901
[4] DEAL R. B.: Problem 4 999. Amer. Math. Monthly 68, 1961, 1010. Solution in Amer. Math. Monthly 70, 1963, 218-219.
[5] GREKOS G.: Répartition des densités des sous-suites d’une suite d’entiers. J. Number Theory 10, 1978, 177-191. · Zbl 0388.10033
[6] KURATOWSKI C.: Topologie I. PWN, Warszawa 1958. · Zbl 0078.14603
[7] MAHARAM D.: The representation of abstract integrals. Trans. Amer. Math. Soc. 75, 1953, 154-184. · Zbl 0051.29203
[8] MAHARAM D.: Finitely additive measures on the integers. Sankhya: Indian J. Stat. 38 A, 1976, 44-59. · Zbl 0383.60008
[9] NIVEN I.: Irrational Numbers. Wiley, New York 1956. · Zbl 0070.27101
[10] OSTMANN H. H.: Additive Zahlentheorie I. Springer-Verlag, Berlin-Göttingen-Heidelberg 1956. · Zbl 0072.03101
[11] ŠALÁT T.: A remark on normal numbers. Rev. Roum. Math. Pures et Appl. XI, 1966, 53-56. · Zbl 0178.38001
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