Fine asymptotic densities for sets of natural numbers

Di Nasso, Mauro

doi:10.1090/S0002-9939-10-10351-7

Fine asymptotic densities for sets of natural numbers
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by Mauro Di Nasso PDF

Proc. Amer. Math. Soc. 138 (2010), 2657-2665 Request permission

Abstract:

By allowing values in non-Archimedean extensions of the unit interval, we consider finitely additive measures that generalize the asymptotic density. The existence of a natural class of such “fine densities” is independent of ZFC.

References

A. Blass, R. Frankiewicz, G. Plebanek, and C. Ryll-Nardzewski, A note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320. MR 1845008, DOI 10.1090/S0002-9939-01-05941-X
R. Creighton Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335–346. MR 54000, DOI 10.2307/2372456
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
Vieri Benci and Mauro Di Nasso, Numerosities of labelled sets: a new way of counting, Adv. Math. 173 (2003), no. 1, 50–67. MR 1954455, DOI 10.1016/S0001-8708(02)00012-9
M. Di Nasso and M. Forti, Numerosities of point sets over the real line, Trans. Amer. Math. Soc., to appear.
M. Di Nasso and M. Forti, Special ultrafilters and asymptotic equinumerosities, in preparation.
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Peter J. Hammond, Non-Archimedean subjective probabilities in decision theory and games, Math. Social Sci. 38 (1999), no. 2, 139–156. MR 1706001, DOI 10.1016/S0165-4896(99)00011-6
H. Halberstam and K. F. Roth, Sequences. Vol. I, Clarendon Press, Oxford, 1966. MR 0210679
A. Yu. Khrennikov, Generalized probabilities taking values in non-Archimedean fields and in topological groups, Russ. J. Math. Phys. 14 (2007), no. 2, 142–159. MR 2318826, DOI 10.1134/S1061920807020033
Dorothy Maharam, Finitely additive measures on the integers, Sankhyā Ser. A 38 (1976), no. 1, 44–59. MR 473132
Alan H. Mekler, Finitely additive measures on ${\rm \textbf {N}}$ and the additive property, Proc. Amer. Math. Soc. 92 (1984), no. 3, 439–444. MR 759670, DOI 10.1090/S0002-9939-1984-0759670-1
Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996. Inverse problems and the geometry of sumsets. MR 1477155, DOI 10.1007/978-1-4757-3845-2
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Edward L. Wimmers, The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877, DOI 10.1007/BF02761683