{"paper":{"title":"Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yunlei Wang","submitted_at":"2026-08-03T04:07:59Z","abstract_excerpt":"We study sparse supports of eigenfunctions on the standard lattice $\\mathbb{Z}^d$. For every $d\\ge3$, any real harmonic function with $u(0)\\ne0$ satisfies \\[\n  |\\mathrm{supp}(u)\\cap Q_n^{(d)}|\\ge \\frac{10^{-10}}{d}\\,n^2\n  \\qquad(n\\ge1). \\] The order $n^2$ is sharp in dimension three. In the zero-potential case, this removes the logarithmic loss in the support-count estimate of Li and Zhang [Duke Math. J. 171 (2022), 327--415]. In high dimensions, our support-only estimates improve Krymskii's support-dimension bound [arXiv:2401.02800], yielding exponents that exceed two for $d\\ge17$ and approac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01673","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.01673/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}