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  1. On expansions of.Quentin Lambotte & Françoise Point - 2020 - Annals of Pure and Applied Logic 171 (8):102809.
    Call a (strictly increasing) sequence (rn) of natural numbers regular if it satisfies the following condition: rn+1/rn→θ∈R>1∪{∞} and, if θ is algebraic, then (rn) satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that (Z,+,0,R) is superstable whenever R is enumerated by a regular sequence. We give two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. We (...)
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  • Decidability and classification of the theory of integers with primes.Itay Kaplan & Saharon Shelah - 2017 - Journal of Symbolic Logic 82 (3):1041-1050.
    We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th which is known to be undecidable by the works of Jockusch, Bateman, and Woods.
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  • Expansions of the group of integers by Beatty sequences.Ayhan Günaydın & Melissa Özsahakyan - 2022 - Annals of Pure and Applied Logic 173 (3):103062.
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  • Weakly minimal groups with a new predicate.Gabriel Conant & Michael C. Laskowski - 2020 - Journal of Mathematical Logic 20 (2):2050011.
    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 AM on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with (...)
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  • Definable groups in dense pairs of geometric structures.Alexander Berenstein & Evgueni Vassiliev - 2022 - Archive for Mathematical Logic 61 (3):345-372.
    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 definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-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, (...)
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  • A new dp-minimal expansion of the integers.Eran Alouf & Christian D’elbée - 2019 - Journal of Symbolic Logic 84 (2):632-663.
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