{
  "abstract": "For a bounded sequence $y=(y_n)\\subset E^*$ and $p>0$, put $G_p(y)=\\{x\\in E:(y_n(x))\\in\\ell_p\\}$ and call $y$ $p$-strongly normal when $G_p(y)$ is dense in $E$. We study the density spectra of the original sequence, its subsequences, and its normalized successive blocks. The possible subsequence and block spectra are exactly $\\varnothing$, $(0,\\infty)$, $(1,\\infty)$, $[r,\\infty)$, and $(r,\\infty)$ for $r>1$, and their possible pairs are precisely the nested pairs. Every such pair has a normalized weakly null $1$-unconditional basic realization in $\\ell_1(\\mathfrak c)^*$, even with empty original-sequence spectrum. Without a basicness requirement, the possible original-sequence spectra are all upper subsets of $(0,\\infty)$; each is realized in a separable Hilbert space. We determine the exact universal perturbation classes: $\\ell_p$ for a fixed original-sequence exponent, $c_0$ for the subsequence spectrum, and $\\ell_1$ for the block spectrum. The ideal-valued version holds for every proper solid permutation-invariant linear ideal $c_{00}\\subset A\\subsetneq c_0$, without a norm or completeness assumption. For the author's 2012 selection model, the bounded-run criterion extends to all proper symmetric quasi-Banach sequence ideals. In the unbounded-run case, a single norm-one projection annihilates the summability domains of any family of fewer than $\\mathfrak b$ such ideals; the quotient by their closed linear span contains a $1$-complemented isometric copy of $\\ell_\\infty$. Such copies also occur in the left jump at every $q>1$ and in the right jump at every $q\\ge1$ of the scale of closed summability domains. Thus the common density criterion coexists with two-sided complemented discontinuities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38853",
  "author_total": 1,
  "authors": [
    {
      "name": "Wiesław Śliwa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f3wass8hh68j7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38853",
  "fulltext_url": null,
  "id": "f3wass8hh68j7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38853",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f3wass8hh68j7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Selection spectra and sharp perturbations of normal sequences",
  "title_text": "Selection spectra and sharp perturbations of normal sequences",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/f3wass8hh68j7",
  "venue": "arXiv",
  "versions": []
}