{
  "abstract": "For $N\\ge3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on $\\mathbb Z^N$ with $u(0)=0$, and set $A_N=(N-2)^2/4$. We prove that the continuum coefficient remains an upper bound, $C(N)\\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1/4$ and $C(4)=A_4=1$. In higher dimensions we show $C(N)<A_N$ for $N=9,10$ and obtain the explicit bound \\[ C(N)\\le 3N-\\sqrt{N^2+8N-8}<2N\\qquad(N\\ge3), \\] which lies below $A_N$ from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh--Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30831",
  "author_total": 1,
  "authors": [
    {
      "name": "Carlos Lizama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x6zqjfv6xffs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30831",
  "fulltext_url": null,
  "id": "0x6zqjfv6xffs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30831",
  "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|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204909495",
  "pdf_url": "https://arxiv.org/pdf/2608.30831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x6zqjfv6xffs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine",
  "title_text": "Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/0x6zqjfv6xffs",
  "venue": "arXiv",
  "versions": []
}