{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:46Z","timestamp":1753893826935,"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 investigate the lower asymptotic density of sumsets in $\\mathbb{N}^2$ by proving certain Pl\u00fcnnecke type inequalities for various notions of lower density in $\\mathbb{N}^2$. More specifically, we introduce a notion of lower tableaux density in $\\mathbb{N}^2$ which involves averaging over convex tableaux-shaped regions in $\\mathbb{N}^2$ which contain the origin. This generalizes the well known Pl\u00fcnnecke type inequality for the lower asymptotic density of sumsets in $\\mathbb{N}$. We also provide a conjectural Pl\u00fcnnecke inequality for the more basic notion of lower rectangular asymtpotic density in $\\mathbb{N}^2$ and prove certain partial results.<\/jats:p>","DOI":"10.37236\/6221","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:49:56Z","timestamp":1578653396000},"source":"Crossref","is-referenced-by-count":0,"title":["Multidimensional Lower Density Versions of Pl\u00fcnnecke\u2019s Inequality"],"prefix":"10.37236","volume":"24","author":[{"given":"Kamil","family":"Bulinski","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:49:59Z","timestamp":1579218599000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,25]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6221","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,8,25]]},"article-number":"P3.34"}}