{"paper":{"title":"On exponential frames near the critical density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Jordy Timo van Velthoven, Marcin Bownik","submitted_at":"2024-11-29T09:15:00Z","abstract_excerpt":"Given a relatively compact set $\\Omega \\subseteq \\mathbb{R}$ of Lebesgue measure $|\\Omega|$ and $\\varepsilon > 0$, we show the existence of a set $\\Lambda \\subseteq \\mathbb{R}$ of uniform density $D (\\Lambda) \\leq (1+\\varepsilon) |\\Omega|$ such that the exponential system $\\{ \\exp(2\\pi i \\lambda \\cdot) \\mathbf{1}_{\\Omega}: \\lambda \\in \\Lambda \\}$ is a frame for $L^2 (\\Omega)$ with frame bounds $A |\\Omega|, B |\\Omega|$ for constants $A,B$ only depending on $\\varepsilon$. This solves a problem on the frame bounds of an exponential frame near the critical density posed by Nitzan, Olevskii and Ula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.19562","kind":"arxiv","version":3},"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/2411.19562/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"}