Journal of Symbolic Logic 87 (1):127-158 (2022)

Abstract
Fix an abelian group $\Gamma $ and an injective endomorphism $F\colon \Gamma \to \Gamma $. Improving on the results of [2], new characterizations are here obtained for the existence of spanning sets, F-automaticity, and F-sparsity. The model theoretic status of these sets is also investigated, culminating with a combinatorial description of the F-sparse sets that are stable in $$, and a proof that the expansion of $$ by any F-sparse set is NIP. These methods are also used to show for prime $p\ge 7$ that the expansion of $$ by multiplication restricted to $t^{\mathbb {N}}$ is NIP.
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DOI 10.1017/jsl.2021.38
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Weakly Minimal Groups with a New Predicate.Gabriel Conant & Michael C. Laskowski - 2020 - Journal of Mathematical Logic 20 (2):2050011.

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