{"paper":{"title":"A Dichotomy for $k$-automatic expansions of Presburger Arithmetic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.FL"],"primary_cat":"math.LO","authors_text":"Alexi Block Gorman, Chris Schulz, Jason Bell","submitted_at":"2025-08-06T19:55:28Z","abstract_excerpt":"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 $(\\mathbb{N},+,k^{\\mathbb{N}})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.04851","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}