{"paper":{"title":"Sub-linear Upper Bounds on Fourier dimension of Boolean Functions in terms of Fourier sparsity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Swagato Sanyal","submitted_at":"2014-07-13T19:05:53Z","abstract_excerpt":"We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\\left(s^{2/3}\\right)$. Our proof method yields an improved bound of $\\widetilde{O}(\\sqrt{s})$ assuming a conjecture of Tsang~\\etal~\\cite{tsang}, that for every Boolean function of sparsity $s$ there is an affine subspace of $\\mathbb{F}_2^n$ of co-dimension $O(\\poly\\log s)$ restricted to which the function is constant. This conjectured bound is tight upto poly-logarithmic factors as the Fourier dimension and sparsity of the address function are quadratically separated. We obtain these bounds by ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3500","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"}