{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:59Z","timestamp":1753893839150,"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>We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S \\subseteq \\mathbb{Z}_q^n$ containing $|S| = \\mu \\cdot q^n$ elements must contain at least $\\delta(q, \\mu) \\cdot q^n \\cdot 2^n$ arithmetic progressions $x, x+d, \\ldots, x+(q-1)\\cdot d$ such that the difference $d$ is restricted to lie in $\\{0,1\\}^n$. Second, we show that for prime $p$ a symmetric set $S\\subseteq\\mathbb{F}_p^n$ with $|S|=\\mu\\cdot p^n$ elements contains at least $\\mu^{C(p)}\\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.<\/jats:p>","DOI":"10.37236\/9242","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T09:40:58Z","timestamp":1600422058000},"source":"Crossref","is-referenced-by-count":0,"title":["On Arithmetic Progressions in Symmetric Sets in Finite Field Model"],"prefix":"10.37236","volume":"27","author":[{"given":"Jan","family":"H\u0105z\u0142a","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,9,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p61\/8180","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p61\/8180","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T09:40:58Z","timestamp":1600422058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,18]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9242","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,9,18]]},"article-number":"P3.61"}}