{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830836,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\Omega = \\bigoplus_{i=1}^\\infty \\mathbb{Z}_3$ and $e_i = (0, \\dots, 0 , 1, 0, \\dots)$ where the $1$ occurs in the $i$-th coordinate. Let $\\mathscr{F}=\\{ \\alpha \\subset \\mathbb{N} : \\varnothing \\neq \\alpha, \\alpha \\text{ is finite} \\}$.\u00a0There is a natural inclusion of $\\mathscr{F}$ into $\\Omega$ where $\\alpha \\in \\mathscr{F}$ is mapped to $e_\\alpha = \\sum_{i \\in \\alpha} e_i$. We give a new proof that if $E \\subset \\Omega$ with $d^*(E) &gt;0$ then there exist $\\omega \\in \\Omega$ and $\\alpha \\in \\mathscr{F}$ such that \\[\u00a0\\{ \\omega, \\omega+ e_\\alpha, \\omega + 2 e_\\alpha \\} \\subset E.\\]Our proof establishes that for the ergodic reformulation of the problem there is a characteristic factor that is a one step compact extension of the Kronecker factor.<\/jats:p>","DOI":"10.37236\/3715","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:03:29Z","timestamp":1578704609000},"source":"Crossref","is-referenced-by-count":0,"title":["A Characteristic Factor for the 3-Term IP Roth Theorem in $\\mathbb{Z}_3^\\mathbb{N}$"],"prefix":"10.37236","volume":"21","author":[{"given":"Randall","family":"McCutcheon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alistair","family":"Windsor","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,7,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:55:20Z","timestamp":1579258520000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i3p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,3]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,7,3]]}},"URL":"https:\/\/doi.org\/10.37236\/3715","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,7,3]]},"article-number":"P3.3"}}