{"paper":{"title":"On the sum of the L1 influences of bounded functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Hamed Hatami, Nathan Keller, Noam Lifshitz, Yuval Filmus","submitted_at":"2014-04-13T15:57:37Z","abstract_excerpt":"Let $f\\colon \\{-1,1\\}^n \\to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Ba\\v{c}kurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 \\pi)+o(d)$ for monot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3396","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":""},"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"}