Abstract
A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.
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References
P. Erdös and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, L'Enseignment Mathématique, Université de Genève, 1980.
H. Halberstam and K. F. Roth, Sequences, Oxford University Press, London, 1966.
C. W. Henson, C. W. M. Kaufmann and H. J. Keisler, The strength of nonstandard methods in arithmetic, Journal of Symbolic Logic 49 (1984), pp. 1039–1058.
G. Kreisel, Axiomatizations of nonstandard analysis that are conservative extensions of formal systems for classical analysis, in: Applications of Model Theory to Algebra, Analysis and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart and Winston, New York, 1969.
S. Leth. Sequences in countable nonstandard models of the natural numbers, this volume.
S. Leth, Applications of nonstandard models and Lebesgue Measure to sequences of natural numbers, to appear in the Transactions of the American Mathematical Society, April 1988.
I. Z. Ruzsa, On difference sets, Studia Scieutiarum Mathematicarum Hungarica 13 (1978), pp. 319–326.
C. L. Stewart and R. Tijdeman, On infinite difference sets, Canadian Journal of Mathematics 31 (1979), pp. 897–910.
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The author wishes to thank the logic group at the University of Wisconsin, and especially Professor Keisler, for their generous support.
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Leth, S.C. Some nonstandard methods in combinatorial number theory. Stud Logica 47, 265–278 (1988). https://doi.org/10.1007/BF00370556
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DOI: https://doi.org/10.1007/BF00370556