{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,11,12]],"date-time":"2024-11-12T05:09:49Z","timestamp":1731388189004,"version":"3.28.0"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T00:00:00Z","timestamp":1716940800000},"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":[[2024,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A linear equation <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline1.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is said to be <jats:italic>sparse<\/jats:italic> if there is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline2.png\"\/><jats:tex-math>\n$c\\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> so that every subset of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline3.png\"\/><jats:tex-math>\n$[n]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of size <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline4.png\"\/><jats:tex-math>\n$n^{1-c}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> contains a solution of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline5.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline6.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline7.png\"\/><jats:tex-math>\n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> variables is <jats:italic>abundant<\/jats:italic> if every subset of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline8.png\"\/><jats:tex-math>\n$[n]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of size <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline9.png\"\/><jats:tex-math>\n$\\varepsilon n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> contains at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline10.png\"\/><jats:tex-math>\n$\\text{poly}(\\varepsilon )\\cdot n^{k-1}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> solutions of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline11.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. It is clear that every abundant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline12.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is sparse, and Gir\u00e3o, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832400018X_inline13.png\"\/><jats:tex-math>\n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in four variables. We further discuss a generalisation of this problem which applies to all linear equations.<\/jats:p>","DOI":"10.1017\/s096354832400018x","type":"journal-article","created":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T09:08:39Z","timestamp":1716973719000},"page":"724-728","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A generalisation of Varnavides\u2019s theorem"],"prefix":"10.1017","volume":"33","author":[{"given":"Asaf","family":"Shapira","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"key":"S096354832400018X_ref3","unstructured":"[3] Bukh, B. (to appear) Extremal graphs without exponentially-small bicliques. Duke Math. J."},{"key":"S096354832400018X_ref7","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS57990.2023.00059"},{"key":"S096354832400018X_ref6","unstructured":"[6] Gir\u00e3o, A. , Hurley, E. , Illingworth, F. and Michel, L. Abundance: asymmetric graph removal lemmas and integer solutions to linear equations. arXiv: 2310.18202."},{"key":"S096354832400018X_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/fms.2016.2"},{"key":"S096354832400018X_ref1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.32.12.331"},{"key":"S096354832400018X_ref11","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-34.3.358"},{"key":"S096354832400018X_ref9","doi-asserted-by":"publisher","DOI":"10.4064\/aa-65-3-259-282"},{"key":"S096354832400018X_ref8","unstructured":"[8] Kosciuszko, T. Counting solutions to invariant equations in dense sets. arXiv: 2306.08567."},{"key":"S096354832400018X_ref4","doi-asserted-by":"publisher","DOI":"10.1112\/jlms.12142"},{"key":"S096354832400018X_ref2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/bds045"},{"key":"S096354832400018X_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579292"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832400018X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,11]],"date-time":"2024-11-11T13:25:03Z","timestamp":1731331503000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832400018X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,29]]},"references-count":11,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["S096354832400018X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832400018x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2024,5,29]]},"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"}}]}}