{"paper":{"title":"On Approximability of Satisfiable $k$-CSPs: VII","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CC","authors_text":"Amey Bhangale, Dor Minzer, Subhash Khot, Yang P. Liu","submitted_at":"2024-11-22T18:58:08Z","abstract_excerpt":"Let $\\Sigma_1,\\ldots,\\Sigma_k$ be finite alphabets, and let $\\mu$ be a distribution over $\\Sigma_1 \\times \\dots \\times \\Sigma_k$ in which the probability of each atom is at least $\\alpha$. We prove that if $\\mu$ does not admit Abelian embeddings, and $f_i: \\Sigma_i \\to \\mathbb{C}$ are $1$-bounded functions (for $i=1,\\ldots,k$) such that \\[ \\left|\\mathbb{E}_{(x_1,\\dots,x_k) \\sim \\mu^{\\otimes n}}\\Big[f_1(x_1) \\dots f_k(x_k)\\Big]\\right| \\geq \\varepsilon, \\] then there exists $L\\colon \\Sigma_1^n\\to\\mathbb{C}$ of degree at most $d$ and $\\|L\\|_2\\leq 1$ such that $|\\langle f_1, L\\rangle|\\geq \\delta$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15136","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15136/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"}