{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T11:18:59Z","timestamp":1753183139648,"version":"3.37.3"},"reference-count":15,"publisher":"Oxford University Press (OUP)","issue":"14","license":[{"start":{"date-parts":[[2017,11,3]],"date-time":"2017-11-03T00:00:00Z","timestamp":1509667200000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Leverhulme Trust Research Fellowship","award":["RF-2016-500"],"award-info":[{"award-number":["RF-2016-500"]}]},{"DOI":"10.13039\/501100000740","name":"University of St Andrews","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000740","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Yoshida scholarship foundation"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,22]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We provide estimates for the dimensions of sets in $\\mathbb{R}$ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say $F$ uniformly avoids APs of length $k \\geq 3$ if there is an $\\epsilon&gt;0$ such that one cannot find an AP of length $k$ and gap length $\\Delta&gt;0$ inside the $\\epsilon \\Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\\epsilon$. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a \u201creverse Kakeya problem:\u201d we show that if the dimension of a set in $\\mathbb{R}^d$ is sufficiently large, then it closely approximates APs in every direction.<\/jats:p>","DOI":"10.1093\/imrn\/rnx261","type":"journal-article","created":{"date-parts":[[2017,10,10]],"date-time":"2017-10-10T11:09:43Z","timestamp":1507633783000},"page":"4419-4430","source":"Crossref","is-referenced-by-count":4,"title":["Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions"],"prefix":"10.1093","volume":"2019","author":[{"given":"Jonathan M","family":"Fraser","sequence":"first","affiliation":[{"name":"School of Mathematics & Statistics, University of St Andrews, St Andrews, UK"}]},{"given":"Kota","family":"Saito","sequence":"additional","affiliation":[{"name":"Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, Japan"}]},{"given":"Han","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics & Statistics, University of St Andrews, St Andrews, UK"}]}],"member":"286","published-online":{"date-parts":[[2017,11,2]]},"reference":[{"key":"2019072205400467800_B1","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1007\/s000390050087","article-title":"\u201cOn the dimension of Kakeya sets and related maximal inequalities.\u201d","volume":"9","author":"Bourgain","year":"1999","journal-title":"Geom. 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Combin.","author":"O\u2019Bryant","year":"2011"},{"volume-title":"Dimensions, Embeddings, and Attractors","year":"2011","author":"Robinson","key":"2019072205400467800_B12"},{"key":"2019072205400467800_B13","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1073\/pnas.28.12.561","article-title":"\u201cOn sets of integers which contain no three terms in arithmetical progression.\u201d","volume":"28","author":"Salem","year":"1942","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"2019072205400467800_B14","first-page":"1929","article-title":"\u201cSalem sets with no arithmetic progressions.\u201d","author":"Shmerkin","year":"2017","journal-title":"Int. Math. Res. 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