{"paper":{"title":"Cayley Digraphs Associated to Arithmetic Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"David Covert, Jonathan Pakianathan, Ye\\c{s}im Demiro\\u{g}lu Karabulut","submitted_at":"2018-08-20T19:30:08Z","abstract_excerpt":"We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-S\\'{a}rk\\\"{o}zy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\\mathbb{F}_q), d \\geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06665","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":""},"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"}