{"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":1753893830548,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For sufficiently large subsets $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{D}$ of $\\mathbb{F}_q$, Gyarmati and S\u00e1rk\u00f6zy (2008)\u00a0 showed the solvability of the equations $a + b= c d$ and $a b + 1 = c d$ with $a \\in \\mathcal{A}$, $b \\in\\mathcal{B}$, $c \\in \\mathcal{C}$, $d \\in \\mathcal{D}$. They asked whether one can extend these results to every $k \\in \\mathbb{N}$ in the following way: for large subsets $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}, \\mathcal{D}$ of $\\mathbb{F}_q$,\u00a0 there are $a_1, \\ldots, a_k, a_1', \\ldots, a_k' \\in\\mathcal{A}$, $b_1, \\ldots, b_k, b_1', \\ldots, b_k' \\in \\mathcal{B}$ with $a_i + b_j, a_i' b_j' + 1 \\in \\mathcal{C}\\mathcal{D}$ (for $1 \\leq i, j\\leq k)$. The author (2010) gave an affirmative answer to this question using Fourier analytic methods. In this paper, we will extend this result to the setting of finite cyclic rings using tools from spectral graph theory.<\/jats:p>","DOI":"10.37236\/2385","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:27:41Z","timestamp":1578713261000},"source":"Crossref","is-referenced-by-count":3,"title":["Sum and Shifted-Product Subsets of Product-Sets over Finite Rings"],"prefix":"10.37236","volume":"19","author":[{"given":"Anh Vinh","family":"Le","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,6,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i2p33\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i2p33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:33:59Z","timestamp":1579300439000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i2p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6,6]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/2385","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,6,6]]},"article-number":"P33"}}