{"paper":{"title":"Quantum and classical low-degree learning via a dimension-free Remez inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.PR","quant-ph"],"primary_cat":"math.AP","authors_text":"Alexander Volberg, Haonan Zhang, Joseph Slote, Ohad Klein","submitted_at":"2023-01-04T04:15:40Z","abstract_excerpt":"Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice $\\binom{[n]}{k}$, the hypergrid $[K]^n$, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $\\{\\exp(2\\pi i k/K)\\}_{k=1}^K$ controls the supremum of $f$ over the entire polytorus $(\\{z\\in\\mathbf{C}:|z|=1\\}^n)$, with multiplicative constant $C$ depending on $K$ and $\\text{deg}(f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.01438","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2301.01438/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}