{"paper":{"title":"On generalized corners and matrix multiplication","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Kevin Pratt","submitted_at":"2023-09-07T17:41:56Z","abstract_excerpt":"Suppose that $S \\subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\\delta), (x+\\delta,y')$, where $\\delta \\neq 0$. How big can $S$ be? Trivially, $n \\le |S| \\le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \\le O(n^2/(\\log \\log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \\ge \\Omega(n \\log n/\\sqrt{\\log \\log n})$.\n  Could it be that for all $\\varepsilon > 0$, $|S| \\le O(n^{1+\\varepsilon})$? We show that if so, this would rule out obtaining"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.03878","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/2309.03878/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"}