{"paper":{"title":"Twisted patterns in large subsets of $\\mathbb{Z}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Kamil Bulinski, Michael Bj\\\"orklund","submitted_at":"2015-12-06T01:15:35Z","abstract_excerpt":"Let $E \\subset \\mathbb{Z}^N$ be a set of positive upper Banach density and let $\\Gamma < \\operatorname{GL}_N(\\mathbb{Z})$ be a finitely generated, strongly irreducible subgroup whose Zariski closure in $\\operatorname{GL}_N(\\mathbb{R})$ is a Zariski connected semisimple group with no compact factors. Let $Y$ be any set and suppose that $\\Psi : \\mathbb{Z}^N \\rightarrow Y$ is a $\\Gamma$-invariant function. We prove that for every positive integer $m$, there exists a positive integer $k$ with the property that for every finite set $F \\subset \\mathbb{Z}^N$ with $|F| = m$, we have \\[ \\Psi(kF) \\subse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01719","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"}