Abstract
The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer d and a subset A ⊆ ℤ whose set of representations base d is recognized by a finite automaton, a necessary condition is found for x + y ∈ A to be a stable formula in Th(ℤ, +, A). Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the d-automatic A ⊆ ℤ such that (ℤ, +, A) is stable. This characterization is in terms of what were called F-sets in [16] and elementary p-nested sets in [10]. Automata-theoretic methods are also used to produce some NIP expansions of (ℤ, +), in particular the expansion by the monoid (dℕ, ×).
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Acknowledgements
I am very grateful to Gabriel Conant, who, upon viewing an earlier draft of this paper, pointed out to me that the cases dealt with in Theorems 3.1 and 4.2 were sufficient to prove the general case. I am also grateful to Jason Bell, in conversations with whom the main theorem was first articulated as a conjecture. Many thanks to the reviewer for their thorough reading and thoughtful feedback; I am in particular in their debt for simplifying the fourth equivalent statement of Corollary 5.2. Finally, I am deeply grateful to my advisor, Rahim Moosa, for excellent guidance, thorough editing, and many helpful discussions.
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This work was partially supported by an NSERC PGS-D and an NSERC CGS-D.
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Hawthorne, C. Automata and tame expansions of (ℤ, +). Isr. J. Math. 249, 651–693 (2022). https://doi.org/10.1007/s11856-022-2322-6
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DOI: https://doi.org/10.1007/s11856-022-2322-6