{
  "abstract": "We study how often $n$ distinct integer dilates of a periodic function can all lie above a fixed threshold or all lie below its negative. We consider continuous functions $f:\\mathbb R\\to[-1,1]$ that are $1$-periodic, even, strictly decreasing on $[0,1/2]$, and satisfy $f(0)=1$ and $f(t+1/2)=-f(t)$. This class includes the cosine and the triangle wave. For $0\\leqα<1$, let $ρ_α\\in(0,1/4]$ be determined by $f(ρ_α)=α$, and let $F_α(n)$ denote the infimum of these proportions over all sets of $n$ distinct positive integer frequencies. We prove that $F_α(n)\\geq 4ρ_α^n$ and that the limit $\\lim_{n\\to\\infty}F_α(n)^{1/n}$ exists for every threshold. When $ρ_α^{-1}$ is an integer, we obtain \\begin{align} F_α(n) &\\leq C_αn^{\\log_2((1-ρ_α)/(1-2ρ_α))}ρ_α^n, \\end{align} and hence the limit equals $ρ_α$. Otherwise, the limit belongs to the interval $[ρ_α,\\lfloorρ_α^{-1}\\rfloor^{-1}]$. The upper bound in the reciprocal case uses a cyclic covering construction previously studied in the context of circular graphs. We give a self-contained proof of the corresponding optimal modulus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38135",
  "author_total": 2,
  "authors": [
    {
      "name": "Christina Giannitsi"
    },
    {
      "name": "José Madrid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/228r11te8022n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38135",
  "fulltext_url": null,
  "id": "228r11te8022n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38135",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38135",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/228r11te8022n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38135",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38135"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exponential decay of threshold correlations for symmetric periodic functions",
  "title_text": "Exponential decay of threshold correlations for symmetric periodic functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/228r11te8022n",
  "venue": "arXiv",
  "versions": []
}