{"paper":{"title":"On the Fourier spectrum of functions on Boolean cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Andreas Defant, Antonio P\\'erez, Mieczys{\\l}aw Masty{\\l}o","submitted_at":"2017-06-12T14:50:11Z","abstract_excerpt":"Let $f$ be a real-valued, degree-$d$ Boolean function defined on the $n$-dimensional Boolean cube $\\{\\pm 1\\}^{n}$, and $f(x) = \\sum_{S \\subset \\{1,\\ldots,d\\}} \\widehat{f}(S) \\prod_{k \\in S} x_k$ its Fourier-Walsh expansion. The main result states that there is an absolute constant $C >0$ such that the $\\ell_{2d/(d+1)}$-sum of the Fourier coefficients of $f:\\{\\pm 1\\}^{n} \\rightarrow [-1,1]$ is bounded by $\\leq C^{\\sqrt{d \\log d}}$. It was recently proved that a similar result holds for complex-valued polynomials on the $n$-dimensional poly torus $\\mathbb{T}^n$, but that in contrast to this a re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03670","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"}