Abstract
Let U be an initial segment of \(^*{\mathbb N}\) closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U, which is the intersection of U and an internal subset of \(^*{\mathbb N}\). Suppose A has lower U-density α strictly between 0 and 3/5. We show that either there exists a standard real \(\epsilon\) > 0 and there are sufficiently large x in A such that | (A+A) ∩ [0, 2x]| > (10/3+\(\epsilon\)) | A ∩ [0, x]| or A is a large subset of an arithmetic progression of difference greater than 1 or A is a large subset of the union of two arithmetic progressions with the same difference greater than 2 or A is a large subset of the union of three arithmetic progressions with the same difference greater than 4.
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References
Bihani P, Jin R (2006) Kneser’s theorem for upper Banach density. J Theor Nombres Bordeaux 18:323–343
Freiman GA (2002) Structure theory of set addition. II. Results and problems. In: Halasz G, Lovasz L, Simonovits M, Sós VT (eds) Paul Erdös and his mathematics, vol I. Bolyai Society mathematical studies, vol 11, pp 243–260. Springer, Berlin
Freiman GA (1973) Foundations of a structural theory of set addition. Translations of mathematical monographs, vol 37. American Mathematical Society, Providence, RI
Freiman GA (1960). Inverse problem of additive number theory, IV. On addition of finite sets, II. Uch Zap Elabuz Gos Pedagog Inst 8: 72–116
Goldblatt R (1998). Lectures on the hyperreals: an introduction to nonstandard analysis. Graduate texts in mathematics, vol 188. Springer, Berlin
Halberstam H and Roth KF (1966). Sequences. Oxford University Press, New York
Hamidoune YO and Plagne A (2002). A generalization of Freiman’s 3k − 3 theorem. Acta Arith 103: 147–156
Henson CW (1997). Foundations of nonstandard analysis: a gentle introduction to nonstandard extension. In: Cutland, NJ, Henson, CW and Arkeryd, L (eds) Nonstandard analysis: theory and applications. pp 1–49. Kluwer, Dordrecht
Jin R (2002). Sumset phenomenon. Proc Am Math Soc 130: 855–861
Jin R (2001). Nonstandard methods for upper Banach density problems. J Number Theory 91: 20–38
Jin R (2003). Inverse problem for upper asymptotic density. Trans Am Math Soc 355: 57–78
Jin R (2006). Solution to the inverse problem for upper asymptotic density. J Reine Angew Math 595: 121–166
Jin R (2007) Freiman’s inverse problem with small doubling property. Adv Math (San Diego) DOI 10.1016/j.aim.2007.06.002
Jin R and Keisler HJ (2003). Abelian group with layered tiles and the sumset phenomenon. Trans Am Math Soc 355: 79–97
Leth SC (1988). Applications of nonstandard models and Lebesgue measure to sequences of natural numbers. Trans Am Math Soc 307: 457–468
Leth SC (1988). Sequences in countable nonstandard models of the natural numbers. Studia Log 47: 243–263
Leth SC (1988). Some nonstandard methods in combinatorial number theory. Studia Log 47: 265–278
Leth SC (2006). Near arithmetic progressions in sparse sets. Proc Am Math Soc 134: 1579–1589
Lev V (1995) On the structure of sets of integers with small doubling property. Unpublished manuscripts
Lev V and Smeliansky PY (1995). On addition of two distinct sets of integers. Acta Arith 70: 85–91
Lindstrom T (1988). An invitation to nonstandard analysis. In: Cutland, N (ed) Nonstandard analysis and its application. pp 1–105. Cambridge University Press, New York
Nathanson MB (1996). Additive number theory: inverse problems and the geometry of sumsets. Graduate texts in mathematics, vol 165. Springer, Berlin
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Jin, R. Inverse problem for cuts. Logic and Analysis 1, 61–89 (2007). https://doi.org/10.1007/s11813-007-0002-9
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DOI: https://doi.org/10.1007/s11813-007-0002-9