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arXiv · 2011.12405

Contributions to the theory of $F$-automatic sets

Christopher Hawthorne
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Abstract

Fix an abelian group $\Gamma$ and an injective endomorphism $F \colon \Gamma \to \Gamma$. Improving on the results of Bell and Moosa, 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 $(\Gamma, +)$, and a proof that the expansion of $(\Gamma, +)$ by any $F$-sparse set is NIP. These methods are also used to show for prime $p\ge 7$ that the expansion of $(\mathbb{F}_p[t], +)$ by multiplication restricted to $t^\mathbb{N}$ is NIP.

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Christopher Hawthorne. 2020-11-24. Contributions to the theory of $F$-automatic sets. https://arxiv.org/abs/2011.12405

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