{"paper":{"title":"$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Donggeun Ryou, Minh-Quy Pham, Shengze Duan","submitted_at":"2024-12-27T23:52:25Z","abstract_excerpt":"For $0 < \\alpha \\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\\mathbb{R}^d$ such that $N(E,\\varepsilon) \\lesssim \\varepsilon^{-\\alpha}$ for $0 <\\varepsilon <1$ where $N(E,\\varepsilon)$ is the smallest number of $\\varepsilon$-balls needed to cover $E$. We proved that if $f \\in L^p(\\mathbb{R}^d)$ with \\begin{align*}\n  1 \\leq p\\leq p_\\alpha:=\n  \\begin{cases}\n  \\frac{d^2+d+2\\alpha}{2\\alpha} & d \\geq 3,\n  \\frac{4}{\\alpha} &d =2,\n  \\end{cases}\n  \\end{align*} and $\\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19956","kind":"arxiv","version":2},"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/2412.19956/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"}