{"paper":{"title":"On side lengths of corners in positive density subsets of the Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"Luka Rimani\\'c, Polona Durcik, Vjekoslav Kova\\v{c}","submitted_at":"2016-09-28T19:57:05Z","abstract_excerpt":"We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\\mathbb{R}^d$ to corners in subsets of $\\mathbb{R}^d\\times\\mathbb{R}^d$. More precisely, if $1<p<\\infty$, $p\\neq 2$, and $d$ is large enough, we show that an arbitrary measurable set $A\\subseteq\\mathbb{R}^d\\times\\mathbb{R}^d$ of positive upper Banach density contains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\\ell^p$-norm of the side $s$ attains all sufficiently large real values. Even though we closely follow the basic steps from [3], the proof diverges at the part relying o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.09056","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":""},"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"}