{
  "abstract": "For $n\\ge2$, a good slope of a permutation $f$ of $\\mathbb{Z}_n=\\mathbb{Z}/n\\mathbb{Z}$ is an element $m\\in\\mathbb{Z}_n$ for which $x\\mapsto f(x)+mx$ is a permutation. We call $f$ congruence-preserving if it preserves congruences modulo every divisor of $n$. For odd prime powers $n$, we classify all permutations with more than $n/2$ good slopes. Among permutations that are not congruence-preserving, we determine the maximum number of good slopes for $n=3^e$ with $e\\ge2$, and for $n=25$, $125$ and $625$. Every permutation of $\\mathbb{Z}_n$ has at most $φ(n)$ good slopes, where $φ$ is Euler's totient function. For several families of moduli with at least two distinct prime divisors, we prove a sharp upper bound for the number of good slopes of the permutations with fewer than $φ(n)$ good slopes. These families include $9q$ and $3q^e$ for every prime $q\\ge5$ and every $e\\ge1$. Together with a companion paper, this settles the bound for every odd composite modulus below $105$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04551",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipak K. Bhunia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gh3q0e9wne6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04551",
  "fulltext_url": null,
  "id": "3gh3q0e9wne6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04551",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gh3q0e9wne6q.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Good slopes, congruence preservation, and gaps over residue rings",
  "title_text": "Good slopes, congruence preservation, and gaps over residue rings",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/3gh3q0e9wne6q",
  "venue": "arXiv",
  "versions": []
}