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Sublinear Michaux-Villemaire over (ℕ, +, kN)

  • Chris Schulz
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Abstract

In 1996, Michaux and Villemaire showed that any nontrivial expansion of (ℕ, +) defines new unary sets. We conjecture that the same holds over the structure (ℕ, +, kN), 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 kN. Using this characterization, we prove that if an expansion of (ℕ, +, kN) defines the same unary sets, then every set of sublinear growth definable in the expansion is already definable in (ℕ, +, kN).

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Cite

BibTeXRIS
BibTeX
@misc{schulz2026sublinear,
  title = {{Sublinear Michaux-Villemaire over $(\mathbbN, +, k^\mathbbN)$}},
  author = {Chris Schulz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26409},
  eprint = {2610.26409},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26409},
}
RIS
TY  - UNPB
TI  - Sublinear Michaux-Villemaire over (ℕ, +, k^N)
AU  - Chris Schulz
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26409
UR  - https://arxiv.org/abs/2610.26409
AB  - In 1996, Michaux and Villemaire showed that any nontrivial expansion of (ℕ, +) defines new unary sets. We conjecture that the same holds over the structure (ℕ, +, k^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^N. Using this characterization, we prove that if an expansion of (ℕ, +, k^N) defines the same unary sets, then every set of sublinear growth definable in the expansion is already definable in (ℕ, +, k^N).
ER  -