{"paper":{"title":"Boolean Functions with Minimal Spectral Sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Jevg\\=enijs Vihrovs, Kri\\v{s}j\\=anis Pr\\=usis","submitted_at":"2024-12-20T17:34:06Z","abstract_excerpt":"We show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $\\Theta(\\sqrt{\\log n})$, which is asymptotically minimal. Our main new function combines the Hamming code with the Boolean address function and has $\\lambda(f) = \\sqrt{(1+o(1)) \\log_2 n}$, which is optimal even up to a constant factor. By combining this function with itself in a specific way, we also obtain a family of functions with $\\text{s}_0(f) = (c+o(1)) \\log_2 n$ and $\\text{s}_0(f) = (1-c+o(1)) \\log_2 n$ for any $c \\in [0,1]$. This is an optimal tradeoff for Boolean functions with low s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.16088","kind":"arxiv","version":2},"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/2412.16088/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"}