{
  "abstract": "Let $f: \\mathbb{N} \\rightarrow (0,\\infty)$ be a multiplicative function, and let $ω> 0$. We consider the frequency with which patterns of the form $$ f(n) > ω\\max\\{f(n-1),f(n+1)\\}, \\quad f(n) < \\frac{1}ω\\min\\{f(n-1),f(n+1)\\} $$ occur in the sequence $(f(n))_n$, and give necessary conditions on $f$ so that the set of occurrences of at least one of these patterns has logarithmic density $0$. This includes a complete classification of all such completely multiplicative functions. As a particular application, we characterise (up to prime powers $p^k$, $k \\geq 2$) all multiplicative functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. This answers, in a strong form, two conjectures of A. Sárközy from 2001. \\\\ The proof is based on a natural ``pretentiousness'' dichotomy for additive functions. Key tools in the analysis include an analogue of the Matomäki-Radziwiłłtheorem for additive functions, due to the author, as well as a new bound for the concentration of the gaps $g(n+1)-g(n)$ of an additive function $g$, which may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08424",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander P. Mangerel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b05fd07e87xh3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08424",
  "fulltext_url": null,
  "id": "b05fd07e87xh3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08424",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220948623",
  "pdf_url": "https://arxiv.org/pdf/2610.08424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b05fd07e87xh3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.08424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Sárközy's Local Extrema Conjectures",
  "title_text": "On Sárközy's Local Extrema Conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/b05fd07e87xh3",
  "venue": "arXiv",
  "versions": []
}