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Fine asymptotic densities for sets of natural numbers
Author:
Mauro Di Nasso
Journal:
Proc. Amer. Math. Soc. 138 (2010), 2657-2665
MSC (2010):
Primary 11B05, 03E05; Secondary 11R21
Posted:
April 1, 2010
MathSciNet review:
2644882
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Additional Information
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.
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- A. Blass, R. Frankiewicz, G. Plebanek, and C. Ryll-Nardzewski, A note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129, 3313-3320 (2001). MR 1845008 (2002i:28002)
- 2.
- R.C. Buck, Generalized asymptotic density, Amer. J. Math. 75, 335-346 (1953). MR 0054000 (14:854f)
- 3.
- C.C. Chang and H.J. Keisler, Model Theory (3rd edition), North-Holland, Amsterdam, 1990. MR 1059055 (91c:03026)
- 4.
- V. Benci and M. Di Nasso, Numerosities of labelled sets: a new way of counting, Adv. Math. 173, 50-67 (2003). MR 1954455 (2004b:03065)
- 5.
- M. Di Nasso and M. Forti, Numerosities of point sets over the real line, Trans. Amer. Math. Soc., to appear.
- 6.
- M. Di Nasso and M. Forti, Special ultrafilters and asymptotic equinumerosities, in preparation.
- 7.
- H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. MR 603625 (82j:28010)
- 8.
- P.J. Hammond, Non-Archimedean subjective probabilities in decision theory and games, Mathematical Social Sciences 38, 139-156 (1999). MR 1706001 (2000i:91022)
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- H. Halberstam and K.F. Roth, Sequences, Oxford University Press, 1966. MR 0210679 (35:1565)
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- A. Khrennikov, Generalized probabilities taking values in non-Archimedean fields and in topological groups, Russian J. Math. Phys. 14, 142-159 (2007). MR 2318826 (2008d:60011)
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- D. Maharam, Finitely additive measures on the integers, Sankhya Ser. A 38, 44-59 (1976). MR 0473132 (57:12810)
- 12.
- A.H. Mekler, Finitely additive measures on
 and the additive property, Proc. Amer. Math. Soc. 92, 439-444 (1984). MR 759670 (86j:28003)
- 13.
- M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, Springer, 1996. MR 1477155 (98f:11011)
- 14.
- S. Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer, 1998. MR 1623206 (98m:03002)
- 15.
- E. Wimmers, The Shelah P-point independence theorem, Israel J. Math. 43, 28-48 (1982). MR 728877 (85e:03118)
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Additional Information
Mauro Di Nasso
Affiliation:
Dipartimento di Matematica, Università di Pisa, Pisa, Italy
Email:
dinasso@dm.unipi.it
DOI:
http://dx.doi.org/10.1090/S0002-9939-10-10351-7
PII:
S 0002-9939(10)10351-7
Keywords:
Asymptotic density,
ultrafilter,
non-Archimedean group
Received by editor(s):
August 21, 2009
Received by editor(s) in revised form:
October 10, 2009
Posted:
April 1, 2010
Communicated by:
Julia Knight
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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