{"paper":{"title":"Small gaps in the spectrum of the rectangular billiard","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.NT","math.PR"],"primary_cat":"math.AP","authors_text":"Jean Bourgain, Maksym Radziwi{\\l}{\\l}, Valentin Blomer, Zeev Rudnick","submitted_at":"2016-04-08T18:02:27Z","abstract_excerpt":"We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $\\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\\alpha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02413","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"}