{"paper":{"title":"An optimal uncertainty principle in twelve dimensions via modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Felipe Gon\\c{c}alves, Henry Cohn","submitted_at":"2017-12-12T18:52:49Z","abstract_excerpt":"We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $f \\colon \\mathbb{R}^{12} \\to \\mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $\\widehat{f}$ by $\\widehat{f}(\\xi) = \\int_{\\mathbb{R}^d} f(x)e^{-2\\pi i \\langle x, \\xi\\rangle}\\, dx$, and suppose $\\widehat{f}$ is real-valued and integrable. We show that if $f(0) \\le 0$, $\\widehat{f}(0) \\le 0$, $f(x) \\ge 0$ for $|x| \\ge r_1$, and $\\widehat{f}(\\xi) \\ge 0$ for $|\\xi| \\ge r_2$, then $r_1r_2 \\ge 2$, and this bound is sharp. The constru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04438","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":""},"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"}