Illinois Journal of Mathematics

Multiplicative structure in stable expansions of the group of integers

Gabriel Conant

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We define two families of expansions of $(\mathbb{Z},+)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega $. The first family consists of expansions $(\mathbb{Z},+,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{Z}^{+}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+)$ by all unary predicates of the form $\{q^{n}:n\in \mathbb{N}\}$ for some $q\in \mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq \mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_{n})_{n=0}^{\infty }\subseteq\mathbb{R}^{+}$ such that $\{\frac{\lambda_{n}}{\lambda_{m}}:m\leq n\}$ is closed and discrete.

Article information

Illinois J. Math., Volume 62, Number 1-4 (2018), 341-364.

Received: 24 March 2018
Revised: 26 September 2018
First available in Project Euclid: 13 March 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C45: Classification theory, stability and related concepts [See also 03C48] 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11U09: Model theory [See also 03Cxx]


Conant, Gabriel. Multiplicative structure in stable expansions of the group of integers. Illinois J. Math. 62 (2018), no. 1-4, 341--364. doi:10.1215/ijm/1552442666.

Export citation


  • M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson and S. Starchenko, Vapnik–Chervonenkis density in some theories without the independence property, II, Notre Dame J. Form. Log. 54 (2013), no. 3–4, 311–363.
  • M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson and S. Starchenko, Vapnik–Chervonenkis density in some theories without the independence property, I, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5889–5949.
  • O. Belegradek, Y. Peterzil and F. Wagner, Quasi-o-minimal structures, J. Symbolic Logic 65 (2000), no. 3, 1115–1132.
  • O. Belegradek and B. Zilber, The model theory of the field of reals with a subgroup of the unit circle, J. Lond. Math. Soc. (2) 78 (2008), no. 3, 563–579.
  • A. Bès, Undecidable extensions of Büchi arithmetic and Cobham–Semënov theorem, J. Symbolic Logic 62 (1997), no. 4, 1280–1296.
  • A. Bès, A survey of arithmetical definability, Bull. Belg. Math. Soc. Simon Stevin suppl (2001), 1–54. A tribute to Maurice Boffa.
  • E. Casanovas and M. Ziegler, Stable theories with a new predicate, J. Symbolic Logic 66 (2001), no. 3, 1127–1140.
  • G. Cherlin and F. Point, On extensions of Presburger arithmetic, Proceedings of the fourth Easter conference on model theory (Gross Köris, 1986) (Seminarberichte), vol. 86, Humboldt Univ., Berlin, 1986, pp. 17–34.
  • G. Conant, Stability and sparsity in sets of natural numbers, 2017; available at \arxivurlarXiv:1701.01387.
  • A. Dolich and J. Goodrick, Strong theories of ordered Abelian groups, Fund. Math. 236 (2017), no. 3, 269–296.
  • J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2) 155 (2002), no. 3, 807–836.
  • H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory 1 (1967), 1–49.
  • U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium '85 (Orsay, 1985), Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987.
  • S. Ibuka, H. Kikyo and H. Tanaka, Quantifier elimination for lexicographic products of ordered Abelian groups, Tsukuba J. Math. 33 (2009), no. 1, 95–129.
  • A. A. Ivanov, The structure of superflat graphs, Fund. Math. 143 (1993), no. 2, 107–117.
  • M. Jarden and W. Narkiewicz, On sums of units, Monatsh. Math. 150 (2007), no. 4, 327–332.
  • Q. Lambotte and F. Point, On expansions of $(\mathbf{Z},+,0)$, 2017; available at \arxivurlarXiv:1702.04795.
  • M. C. Laskowski, Mutually algebraic structures and expansions by predicates, J. Symbolic Logic 78 (2013), no. 1, 185–194.
  • D. Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002.
  • C. Michaux and R. Villemaire, Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham's and Semenov's theorems, Ann. Pure Appl. Logic 77 (1996), no. 3, 251–277.
  • J. C. M. Nash and M. B. Nathanson, Cofinite subsets of asymptotic bases for the positive integers, J. Number Theory 20 (1985), no. 3, 363–372.
  • D. Palacín and R. Sklinos, On superstable expansions of free Abelian groups, Notre Dame J. Form. Log. 59 (2018), no. 2, 157–169.
  • A. Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press Oxford University Press, New York, 1983.
  • A. Pillay, The model-theoretic content of Lang's conjecture, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 101–106.
  • F. Point, On decidable extensions of Presburger arithmetic: From A. Bertrand numeration systems to Pisot numbers, J. Symbolic Logic 65 (2000), no. 3, 1347–1374.
  • B. Poizat, Supergénérix, J. Algebra 404 (2014), 240–270. À la mémoire d'Éric Jaligot. [In memoriam Éric Jaligot].
  • M. Prest, Model theory and modules, handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003.
  • T. Scanlon, A proof of the André–Oort conjecture via mathematical logic [after Pila, Wilkie and Zannier], Astérisque 1037 (2012), no. 348, 299–315. Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042.
  • W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551.
  • A. L. Semenov, On certain extensions of the arithmetic of addition of natural numbers, Math. USSR, Izv. 15 (1980), no. 2, 401–418.
  • R. J. Stroeker and R. Tijdeman, Diophantine equations, computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982.
  • L. van den Dries and A. Günayd\in, The fields of real and complex numbers with a small multiplicative group, Proc. Lond. Math. Soc. (3) 93 (2006), no. 1, 43–81.
  • A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094.