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arXiv · 2508.04851

A Dichotomy for $k$-automatic expansions of Presburger Arithmetic

Jason Bell·Alexi Block Gorman·Chris Schulz
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Abstract

Let $k\ge 2$ and let $X$ be a subset of the natural numbers that is $k$-automatic and not eventually periodic. We show that the following dichotomy holds: either all $k$-automatic subsets are definable in the expansion of Presburger arithmetic in which we adjoin the predicate $X$, or $(\mathbb{N},+,X)$ has the same definable sets as $(\mathbb{N},+,k^{\mathbb{N}})$.

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Jason Bell, Alexi Block Gorman, Chris Schulz. 2025-08-06. A Dichotomy for $k$-automatic expansions of Presburger Arithmetic. https://arxiv.org/abs/2508.04851

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