{"paper":{"title":"Quantitative flatness and obstructions in Fourier analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.DS","math.FA","math.MG"],"primary_cat":"math.CA","authors_text":"Jonathan M. Fraser","submitted_at":"2026-06-11T10:42:23Z","abstract_excerpt":"Three important problems in Fourier analysis are the Fourier restriction problem, the $L^p$-improving problem, and the Fourier decay problem. Positive results for any of these problems require a quantitative understanding of various geometric properties of the given measure, including curvature and arithmetic resonance. In this paper we establish a unified framework for providing negative results for all three problems (that is, we provide explicit obstructions to a measure satisfying certain Fourier restriction, $L^p$-improving, or Fourier decay estimates) by quantifying flat parts of the mea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13170","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/2606.13170/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"}