{"paper":{"title":"Fourier uniformity of bounded multiplicative functions in short intervals on average","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kaisa Matom\\\"aki, Maksym Radziwi{\\l}{\\l}, Terence Tao","submitted_at":"2018-12-04T05:31:07Z","abstract_excerpt":"Let $\\lambda$ denote the Liouville function. We show that as $X \\rightarrow \\infty$, $$ \\int_{X}^{2X} \\sup_{\\alpha} \\left | \\sum_{x < n \\leq x + H} \\lambda(n) e(-\\alpha n) \\right | dx = o ( X H) $$ for all $H \\geq X^{\\theta}$ with $\\theta > 0$ fixed but arbitrarily small. Previously, this was only known for $\\theta > 5/8$. For smaller values of $\\theta$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.01224","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":""},"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"}