{"paper":{"title":"Fourier dimension and spectral gaps for hyperbolic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO","math.DS"],"primary_cat":"math.CA","authors_text":"Jean Bourgain, Semyon Dyatlov","submitted_at":"2017-04-10T15:35:10Z","abstract_excerpt":"We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\\Gamma\\backslash\\mathbb H^2$ which depends only on the dimension $\\delta$ of the limit set. More precisely, we show that when $\\delta>0$ there exists $\\varepsilon_0=\\varepsilon_0(\\delta)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\\Re s>\\delta-\\varepsilon_0$.\n  The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589]. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent intere"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02909","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":""},"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"}