{
  "abstract": "In 1996, Michaux and Villemaire showed that any nontrivial expansion of $(\\mathbb{N}, +)$ defines new unary sets. We conjecture that the same holds over the structure $(\\mathbb{N}, +, k^\\mathbb{N})$, and we prove a partial result in this direction. To do so, we first show that every nonempty sparse $k$-automatic set is a Presburger-definable image of some Cartesian power of $k^\\mathbb{N}$. Using this characterization, we prove that if an expansion of $(\\mathbb{N}, +, k^\\mathbb{N})$ defines the same unary sets, then every set of sublinear growth definable in the expansion is already definable in $(\\mathbb{N}, +, k^\\mathbb{N})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26409",
  "author_total": 1,
  "authors": [
    {
      "name": "Chris Schulz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fetst1bpea6ne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26409",
  "fulltext_url": null,
  "id": "fetst1bpea6ne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26409",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7223048921",
  "pdf_url": "https://arxiv.org/pdf/2610.26409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fetst1bpea6ne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Sublinear Michaux-Villemaire over $(\\mathbb{N}, +, k^\\mathbb{N})$",
  "title_text": "Sublinear Michaux-Villemaire over (ℕ, +, k^N)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fetst1bpea6ne",
  "venue": "arXiv",
  "versions": []
}