{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:57Z","timestamp":1753893837699,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Fix a prime $p \\geq 3$, and a real number $0  &lt;  \\alpha \\leq 1$. Let $S \\subset {\\mathbb F}_p^n$ be any set with the least number of solutions to  $x + y = 2z$ (note that this means that $x,z,y$ is an arithmetic progression),  subject to the constraint that $|S| \\geq \\alpha p^n$.  What can one say about the structure of such sets $S$?  In this paper we show that they are \"essentially\" the union of a small number of cosets of some large-dimensional subspace of  ${\\mathbb F}_p^n$.  <\/jats:p>","DOI":"10.37236\/400","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:09:49Z","timestamp":1578697789000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Structure of Sets with Few Three-Term Arithmetic Progressions"],"prefix":"10.37236","volume":"17","author":[{"given":"Ernie","family":"Croot","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,9,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r128\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r128\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:23:22Z","timestamp":1579285402000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r128"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/400","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,9,22]]},"article-number":"R128"}}