{
  "abstract": "For finite-dimensional Banach spaces $E$ and $F$, set \\[ ρ(E,F):=\\sup_{0\\neq z\\in E\\otimes F}\\frac{π(z)}{\\varepsilon(z)}. \\] The square growth of $ρ(\\ell_p^d,\\ell_q^d)$ was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If $r=\\min\\{n,m\\}$ and $η(t)=\\min\\{1/t,1/t'\\}$, then, whenever $p$ and $q$ lie on the same side of $2$, \\[ ρ(\\ell_p^n,\\ell_q^m)\\asymp_{p,q} r^{1/2}\\min\\{n^{η(p)},m^{η(q)}\\}. \\] Thus the classical square exponent splits into two dimensional scales. In the mixed range $1\\le p\\le2\\le q\\le\\infty$, writing $a=1/p$ and $b=1/q$, we obtain the lower estimate \\[ ρ_{p,q}(n,m)\\gtrsim_{p,q} \\max\\!\\left\\{r^{\\min\\{a+b,2-a-b\\}}, m^{b-1/2}r^{1/2}\\min\\{n^{1-a},m^{1/2}\\}\\right\\}, \\] and the upper estimate \\[ ρ_{p,q}(n,m)\\lesssim_{p,q} \\min\\!\\left\\{n^{1-a}r^{1-b},m^b r^a,r\\right\\}. \\] Moreover, if $(n-m)(a+b-1)\\le0$, then \\[ ρ_{p,q}(n,m)\\asymp_{p,q} r^{\\min\\{a+b,\\,2-a-b\\}}. \\] In the interior mixed range $1<p<2<q<p'$, if \\[ α_{p,q}:= \\frac{(a+b-1)(\\frac12-b)}{a-\\frac12}, \\] then \\[ ρ_{p,q}(n,m)\\lesssim_{p,q}m^{1-α_{p,q}}, \\qquad \\sup_{n\\ge1}ρ_{p,q}(n,m)\\asymp_{p,q}m^{1-α_{p,q}}. \\] The corresponding statements in the reversed mixed range follow by duality and symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04469",
  "author_total": 3,
  "authors": [
    {
      "name": "Daniel Nunez-Alarcon"
    },
    {
      "name": "Daniel Pellegrino"
    },
    {
      "name": "Joedson Santos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6820cn0wzv7mg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04469",
  "fulltext_url": null,
  "id": "6820cn0wzv7mg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04469",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7211920482",
  "pdf_url": "https://arxiv.org/pdf/2609.04469",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6820cn0wzv7mg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.04469",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04469"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The phase diagram of injective-to-projective tensor distortion",
  "title_text": "The phase diagram of injective-to-projective tensor distortion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/6820cn0wzv7mg",
  "venue": "arXiv",
  "versions": []
}