{"paper":{"title":"A blurred view of Van der Waerden type theorems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Marcelo Sales, Vojtech R\\\"odl","submitted_at":"2021-02-09T05:16:37Z","abstract_excerpt":"Let $AP_k=\\{a,a+d,\\ldots,a+(k-1)d\\}$ be an arithmetic progression. For $\\epsilon>0$ we call a set $AP_k(\\epsilon)=\\{x_0,\\ldots,x_{k-1}\\}$ an $\\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\\epsilon d$ holds for all $i\\in\\{0,1\\ldots,k-1\\}$. Complementing earlier results of Dumitrescu, in this paper we study numerical aspects of Van der Waerden, Szemeredi and Furstenberg-Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $\\epsilon$-approximation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.04651","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/2102.04651/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"}