On Superstable Expansions of Free Abelian Groups

Palacín, Daniel; Sklinos, Rizos

doi:10.1215/00294527-2017-0023

Cited by 19 publications

(66 citation statements)

References 10 publications

(10 reference statements)

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“…Then N d s is mutually algebraic for any d ≥ 1 (in fact, it follows from [21] that any structure containing only unary injective functions is mutually algebraic). In each example of a stable expansion of the form (Z, +, A) considered in the sources above, it is shown that A (Z,+) is interpretable in an expansion of N d s by unary predicates, for some d ≥ 1 (in fact, d = 1 suffices for all examples considered in [10], [20], and [26]).…”

Section: Weakly Minimal Abelian Groupsmentioning

confidence: 99%

“…Multiplication of ordinals (denoted ·) extends to Ord ∪ {∞} in the obvious way. The following result is [10, Theorem 2.11], and is proved using Proposition 2.4 and techniques similar to the work of Palacín and Sklinos [26] on the expansion of (Z, +) by {2 n : n ∈ N}. Theorem 2.7 (Conant).…”

Section: Bounded Sets In Weakly Minimal Theoriesmentioning

confidence: 99%

“…Proof. The proof uses techniques similar to those of Palacín and Sklinos [26] and Lambotte and Point [20] (see also Remark 6.4). Let A = {a n } ∞ n=0 be an enumeration of A such that either lim n→∞ an+1 an diverges or converges to a transcendental τ ∈ C, with |τ | > 1.…”

Section: Stable Expansions Of Weakly Minimal Abelian Groupsmentioning

confidence: 99%

“…Motivated by the work on expansions of (Z, +) from [10], [11], [20], and [26], we then focus our attention on abelian groups whose pure theory is weakly minimal (see Proposition 5.1 for an algebraic characterization of such groups). In Proposition 5.4, we observe that if G = (G, +) is a weakly minimal abelian group, and A ⊆ G, then the induced structure A G consists of the quantifier-free induced structure, denoted A qf G , together with unary predicates for A ∩ nG for all n ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

“…The examples in Section 6.1 generalize certain families of "sparse sets" considered in [10], [20], and [26]. In this case, we use methods similar to Lambotte and Point [20] to show that A qf Z is interdefinable with A in the language of equality.…”

Section: Introductionmentioning

confidence: 99%

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Weakly minimal groups with a new predicate

Conant

Laskowski

2020

J. Math. Log.

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Fix a weakly minimal (i.e., superstable U -rank 1) structure M. Let M * be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M . We show that all formulas in the expansion (M * , A) are equivalent to bounded formulas, and so (M, A) is stable (or NIP) if and only if the M-induced structure A M on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with a weakly minimal theory, and A M is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of (Z, +). Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form (M, A). Most notably, we show that if (G, +) is a weakly minimal additive subgroup of the algebraic numbers, A ⊆ G is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of A is a root of unity, then (G, +, B) is superstable for any B ⊆ A.

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Section: Weakly Minimal Abelian Groupsmentioning

confidence: 99%

Section: Bounded Sets In Weakly Minimal Theoriesmentioning

confidence: 99%

Section: Stable Expansions Of Weakly Minimal Abelian Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weakly minimal groups with a new predicate

Conant

Laskowski

2020

J. Math. Log.

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Definable groups in dense pairs of geometric structures

Berenstein¹,

Vassiliev

2021

Arch. Math. Logic

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We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large (in the sense of dimension over the predicate) definable subgroups (the new language) of groups definable in the original language L are also L-definable, and definably amenable L-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T , then the (type-definable) connected component G 00 L P of G in the expansion agrees with the connected component G 00 in the original language. We prove similar preservation results for G ∩ P(M) n , the P-part of G in a lovely pair (M, P), and for subgroups of G in pairs (R, G) where R is a real closed field and G is a subgroup of R >0 or the unit circle S(R) with the Mann property. Keywords Unary predicate expansions • Definable groups • Amenable groups • Mann property • Connected component Mathematics Subject Classification 03C45 • 03C64A. Berenstein was partially supported by Colciencias' Project Teoría de modelos y dinámicas topológicas number 120471250707. E. Vassiliev was partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant AP05134992 Conservative extensions, countable ordered models, and closure operators). Both authors were also supported by the project Expansiones densas de teorías geométricas: grupos y medidas, Facultad de Ciencias, Universidad de los Andes. The authors thank Erik Walsberg for providing the proof for Proposition 4.11.

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Stability and sparsity in sets of natural numbers

Conant

2019

Isr. J. Math.

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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 :). 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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On Superstable Expansions of Free Abelian Groups

Cited by 19 publications

References 10 publications

Weakly minimal groups with a new predicate

Weakly minimal groups with a new predicate

Definable groups in dense pairs of geometric structures

Stability and sparsity in sets of natural numbers

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