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Stability and sparsity in sets of natural numbers

  • Gabriel ConantEmail author
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Abstract

Given a set A ⊆ N, we consider the relationship between stability of the structure (ℤ,+, 0,A) and sparsity of the set A. We first show that a strong enough sparsity assumption on A yields stability of (ℤ, +, 0, A). Specifically, if there is a function f: A → ℝ+ such that supa∈A |af(a)| < ∞ and {\(\frac{s}{t}:s,t \in f(A) \), ts} is closed and discrete, then (ℤ, +, 0, A) is superstable (of U-rank ω if A is infinite). Such sets include examples considered by Palacín and Sklinos [19] and Poizat [23], many classical linear recurrence sequences (e.g., the Fibonaccci numbers), and any set in which the limit of ratios of consecutive elements diverges. Finally, we consider sparsity conclusions on sets A ⊆ N, which follow from model theoretic assumptions on (ℤ, +, 0, A). We use a result of Erdős, Nathanson and Sárközy [8] to show that if (ℤ, +, 0, A) does not define the ordering on ℤ, then the lower asymptotic density of any finitary sumset of A is zero. Finally, in a theorem communicated to us by Goldbring, we use a result of Jin [11] to show that if (ℤ, +, 0,A) is stable, then the upper Banach density of any finitary sumset of A is zero.

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References

  1. [1]
    B. Baizhanov and J. T. Baldwin, Local homogeneity, Journal of Symbolic Logic 69 (2004), 1243–1260.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Baudisch, Magidor–Malitz quantifiers in modules, Journal of Symbolic Logic 49 (1984), 1–8.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    O. Belegradek, Y. Peterzil and F. Wagner, Quasi-o-minimal structures, Journal of Symbolic Logic 65 (2000), 1115–1132.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers, Birkhäuser, Basel, 1992.CrossRefzbMATHGoogle Scholar
  5. [5]
    E. Casanovas and M. Ziegler, Stable theories with a new predicate, Journal of Symbolic Logic 66 (2001), 1127–1140.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Annals of Pure and Applied Logic 95 (1998), 71–92.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    G. Conant, Multiplicative structure in stable expansions of the group of integers, Illinois Journal of Mathematics, to appear, arXiv:1704.00105.Google Scholar
  8. [8]
    P. Erdős, M. B. Nathanson and A. Sárközy, Sumsets containing infinite arithmetic progressions, Journal of Number Theory 28 (1988), 159–166.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Mathematical Surveys and Monographs, Vol. 104, American Mathematical Society, Providence, RI, 2003.Google Scholar
  10. [10]
    D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringsches Problem), Mathematische Annalen 67 (1909), 281–300.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Jin, Nonstandard methods for upper Banach density problems, Journal of Number Theory 91 (2001), 20–38.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E. Kamke, Verallgemeinerungen des Waring–Hilbertschen Satzes, Mathematische Annalen 83 (1921), 85–112.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I. Kaplan and S. Shelah, Decidability and classification of the theory of integers with primes, Journal of Symbolic Logic 82 (2017), 1041–1050.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Q. Lambotte and F. Point, On expansions of (Z,+, 0), arXiv:1702.04795.Google Scholar
  15. [15]
    D. Marker, Model Theory, Graduate Texts in Mathematics, Vol. 217, Springer-Verlag, New York, 2002.Google Scholar
  16. [16]
    E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, AmericanMathematical Monthly 67 (1960), 745–752.Google Scholar
  17. [17]
    J. C. M. Nash and M. B. Nathanson, Cofinite subsets of asymptotic bases for the positive integers, Journal of Number Theory 20 (1985), 363–372.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. B. Nathanson, Additive Number Theory. The Classical Bases, Graduate Texts in Mathematics, Vol. 164, Springer-Verlag, New York, 1996.Google Scholar
  19. [19]
    D. Palacín and R. Sklinos, Superstable expansions of free abelian groups, Notre Dame Journal of Formal Logic 59 (2018), 157–169.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. Pillay, Geometric Stability Theory, Oxford Logic Guides, Vol. 32, The Clarendon Press, Oxford University Press, New York, 1996.Google Scholar
  21. [21]
    B. Poizat, A Course in Model Theory, Universitext, Springer-Verlag, New York, 2000.CrossRefGoogle Scholar
  22. [22]
    B. Poizat, Stable Groups, Mathematical Surveys and Monographs, Vol. 87, American Mathematical Society, Providence, RI, 2001.Google Scholar
  23. [23]
    B. Poizat, Supergénérix, Journal of Algebra 404 (2014), 240–270.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    B. Poonen, Representing numbers in a non-integer base with few (but possibly negative) nonzero digits, MathOverflow, http://mathoverflow.net/q/12177 (version: 2010-01-18).Google Scholar
  25. [25]
    M. Prest, Model theory and modules, in Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 227–253.Google Scholar
  26. [26]
    W. M. Schmidt, Linear recurrence sequences, in Diophantine approximation (Cetraro, 2000), Lecture Notes in Mathematics, Vol. 1819, Springer, Berlin, 2003, pp. 171–247.Google Scholar
  27. [27]
    L. Schnirelmann, Über additive Eigenschaften von Zahlen, Mathematische Annalen 107 (1933), 649–690.MathSciNetGoogle Scholar
  28. [28]
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199–245.MathSciNetCrossRefzbMATHGoogle Scholar

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© Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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