Abstract
In this chapter, we define the notion, due to Leth, of internal subsets of the hypernatural numbers with the Interval Measure Property. Roughly speaking, such sets have a tight relationship between sizes of gaps of the set on intervals and the Lebesgue measure of the image of the set under a natural projection onto the standard unit interval. This leads to a notion of standard subsets of the natural numbers having the Standard Interval Measure Property and we enumerate some basic facts concerning sets with this property.
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Notes
- 1.
This terminology does not appear in the original article of Leth.
- 2.
Here, subprogression means that every block [b′ + id, b′ + id + j] is contained in the corresponding block [b + id, b + id + w].
References
I. Goldbring, S. Leth, On supra-SIM sets of natural numbers. arXiv:1805.05933 (2018)
S. Leth, Some nonstandard methods in combinatorial number theory. Stud. Log. 47(3), 265–278 (1988)
C.L. Stewart, R. Tijdeman, On infinite-difference sets. Can. J. Math. 31(5), 897–910 (1979)
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). The Interval Measure Property. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_15
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DOI: https://doi.org/10.1007/978-3-030-17956-4_15
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