{"paper":{"title":"On commuting pairs in arbitrary sets of 2x2 matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Akshat Mudgal","submitted_at":"2024-11-15T18:19:49Z","abstract_excerpt":"Let $\\textrm{Mat}_2(\\mathbb{R})$ be the set of $2 \\times 2$ matrices with real entries. For any $\\varepsilon>0$ and any finitely--supported probability measure $\\mu$ on $\\textrm{Mat}_2(\\mathbb{R})$, we prove that either \\[ T(\\mu) = \\sum_{X, Y \\in {\\rm supp}(\\mu), XY = YX} \\mu(X) \\mu(Y) < \\varepsilon \\] or there exists some finite set ${S}$ contained in a $2$-dimensional subspace of $\\textrm{Mat}_2(\\mathbb{R})$ such that $\\mu({S}) \\geq \\varepsilon/8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \\[ \\mu ( (a_{i,j})_{1 \\leq i,j \\leq 2} ) = \\nu(a_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.10404","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/2411.10404/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"}