{
  "abstract": "Fix a finite alphabet $\\Sigma$ having at least two letters, and natural numbers $n,k\\geq 1$. We denote by $D_n(\\mathbf{\\Pi}^0_k)$ the class of differences of $n$ subsets of the Cantor space $\\Sigma^\\omega$ in $\\mathbf{\\Pi}^0_k$, and by $D^-_n(\\mathbf{\\Pi}^0_k)$ the class of complements of sets in $D_n(\\mathbf{\\Pi}^0_k)$. We prove, for any natural numbers $n,k\\geq 1$, the existence of a finite alphabet $\\Sigma$ having at least two letters and a finitary language $U_{n,k}$ over $\\Sigma$ accepted by a one-counter automaton such that the $\\omega$-power $$U_{n,k}^\\infty:=\\{ w_0w_1\\cdots\\in\\Sigma^\\omega\\mid\\forall i\\in\\omega ~~w_i\\in U_{n,k}\\}$$ is $D_{2n-1}(\\mathbf{\\Pi}^0_k)$-complete. We also prove a conditional result for the dual classes $D^-_{2n}(\\mathbf{\\Pi}^0_k)$: under the assumption that there exists a finitary language $V_{n,3}$ accepted by a one-counter automaton, such that its $\\omega$-power $V_{n,3}^\\infty$ is $D^-_{2n}(\\mathbf{\\Pi}^0_3)$-complete, there would also exist, for any integer $k>3$, a finitary language $V_{n,k}$ accepted by a one-counter automaton, such that its $\\omega$-power $V_{n,k}^\\infty$ is $D^-_{2n}(\\mathbf{\\Pi}^0_k)$-complete. Finally we add another conditionnal result about Wadge degrees of rational $\\mathbf{\\Delta}^0_2$ omega powers, which completes the recent results of [FGL26].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16473",
  "author_total": 3,
  "authors": [
    {
      "name": "Jacques Duparc"
    },
    {
      "name": "Olivier Finkel"
    },
    {
      "name": "Dominique Lecomte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bs0qn6vxcze8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16473",
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  "id": "1bs0qn6vxcze8",
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  "pdf_url": "https://hal.science/hal-05790654/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bs0qn6vxcze8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:53Z",
      "native_id": "hal-05790654",
      "source": "hal",
      "url": "https://hal.science/hal-05790654v1"
    },
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.16473",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16473"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Some complete ω-powers of a one-counter language in the difference hierarchy",
  "title_text": "Some complete ω-powers of a one-counter language in the difference hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/1bs0qn6vxcze8",
  "venue": "HAL",
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}