{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:58:48Z","timestamp":1747173528015,"version":"3.40.5"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2021,11,26]],"date-time":"2021-11-26T00:00:00Z","timestamp":1637884800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline1.png\"\/><jats:tex-math>\n$\\mathrm{AP}_k=\\{a,a+d,\\ldots,a+(k-1)d\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be an arithmetic progression. For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline2.png\"\/><jats:tex-math>\n$\\varepsilon&gt;0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> we call a set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline3.png\"\/><jats:tex-math>\n$\\mathrm{AP}_k(\\varepsilon)=\\{x_0,\\ldots,x_{k-1}\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline4.png\"\/><jats:tex-math>\n$\\varepsilon$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-approximate arithmetic progression if for some <jats:italic>a<\/jats:italic> and <jats:italic>d<\/jats:italic>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline5.png\"\/><jats:tex-math>\n$|x_i-(a+id)|&lt;\\varepsilon d$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> holds for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline6.png\"\/><jats:tex-math>\n$i\\in\\{0,1\\ldots,k-1\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Complementing earlier results of Dumitrescu (2011, <jats:italic>J. Comput. Geom.<\/jats:italic><jats:bold>2<\/jats:bold>(1) 16\u201329), in this paper we study numerical aspects of Van der Waerden, Szemer\u00e9di and Furstenberg\u2013Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548321000535_inline7.png\"\/><jats:tex-math>\n$\\varepsilon$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-approximation.<\/jats:p>","DOI":"10.1017\/s0963548321000535","type":"journal-article","created":{"date-parts":[[2021,11,26]],"date-time":"2021-11-26T08:31:10Z","timestamp":1637915470000},"page":"684-701","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A blurred view of Van der Waerden type theorems"],"prefix":"10.1017","volume":"31","author":[{"given":"Vojtech","family":"R\u00f6dl","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7561-8073","authenticated-orcid":false,"given":"Marcelo","family":"Sales","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,11,26]]},"reference":[{"doi-asserted-by":"publisher","key":"S0963548321000535_ref17","DOI":"10.1137\/1.9781611975482.27"},{"doi-asserted-by":"publisher","key":"S0963548321000535_ref23","DOI":"10.1002\/rsa.20017"},{"doi-asserted-by":"publisher","key":"S0963548321000535_ref2","DOI":"10.4153\/CMB-1968-047-7"},{"unstructured":"[15] Green, B. and Wolf, J. 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