{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T03:16:08Z","timestamp":1767842168184,"version":"3.49.0"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:00:00Z","timestamp":1709596800000},"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":[[2024,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Tao and Vu showed that every centrally symmetric convex progression <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000051_inline1.png\"\/><jats:tex-math>\n$C\\subset \\mathbb{Z}^d$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is contained in a generalized arithmetic progression of size <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000051_inline2.png\"\/><jats:tex-math>\n$d^{O(d^2)} \\# C$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Berg and Henk improved the size bound to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000051_inline3.png\"\/><jats:tex-math>\n$d^{O(d\\log d)} \\# C$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We obtain the bound <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000051_inline4.png\"\/><jats:tex-math>\n$d^{O(d)} \\# C$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John\u2019s theorem.<\/jats:p>","DOI":"10.1017\/s0963548324000051","type":"journal-article","created":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T07:15:47Z","timestamp":1709622947000},"page":"484-486","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Sharp bounds for a discrete John\u2019s theorem"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2323-2897","authenticated-orcid":false,"given":"Peter","family":"van Hintum","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4605-5045","authenticated-orcid":false,"given":"Peter","family":"Keevash","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,3,5]]},"reference":[{"key":"S0963548324000051_ref3","first-page":"379","article-title":"A counterexample to a strong variant of the Polynomial Freiman-Ruzsa Conjecture in Euclidean space","volume":"8","author":"Lovett","year":"2017","journal-title":"Discrete Anal."},{"key":"S0963548324000051_ref5","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.aim.2008.05.002","article-title":"John-type theorems for generalized arithmetic progressions and iterated sumsets","volume":"219","author":"Tao","year":"2008","journal-title":"Adv. Math."},{"key":"S0963548324000051_ref4","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511755149","volume-title":"Additive Combinatorics","author":"Tao","year":"2006"},{"key":"S0963548324000051_ref2","first-page":"187","volume-title":"Studies and Essays, Presented to R. Courant on his 60th Birthday,","author":"John","year":"1948"},{"key":"S0963548324000051_ref1","doi-asserted-by":"crossref","first-page":"367","DOI":"10.2140\/moscow.2019.8.367","article-title":"Discrete analogues of John\u2019s theorem","volume":"8","author":"Berg","year":"2019","journal-title":"Moscow J. Comb. Number Theory"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548324000051","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T16:08:25Z","timestamp":1728403705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548324000051\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,5]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["S0963548324000051"],"URL":"https:\/\/doi.org\/10.1017\/s0963548324000051","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,5]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}