{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:13Z","timestamp":1753893853887,"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>For a prime number $p$ and a sequence of integers $a_0,\\dots,a_k\\in \\{0,1,\\dots,p\\}$, let $s(a_0,\\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\\dots,x_k)\\in A_0\\times\\dots\\times A_k$ with $x_0=x_1+\\dots + x_k$, over subsets $A_0,\\dots,A_k\\subseteq\\mathbb{Z}_p$ of sizes $a_0,\\dots,a_k$ respectively. We observe that an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length. The same conclusion also holds for the related problem, posed by Bajnok, when $a_0=\\dots=a_k=:a$ and $A_0=\\dots=A_k$, provided $k$ is not equal 1 modulo $p$. Finally, by applying basic Fourier analysis, we show for Bajnok's problem that if $p\\geqslant 13$ and $a\\in\\{3,\\dots,p-3\\}$ are fixed while $k\\equiv 1\\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.\r\n\u00a0\r\nA corrigendum was added March 12, 2019.<\/jats:p>","DOI":"10.37236\/7376","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:15:37Z","timestamp":1578640537000},"source":"Crossref","is-referenced-by-count":0,"title":["Minimum Number of Additive Tuples in Groups of Prime Order"],"prefix":"10.37236","volume":"26","author":[{"given":"Ostap","family":"Chervak","sequence":"first","affiliation":[]},{"given":"Oleg","family":"Pikhurko","sequence":"additional","affiliation":[]},{"given":"Katherine","family":"Staden","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,2,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p30\/7790","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p30\/7790","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:18:10Z","timestamp":1579234690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i1p30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/7376","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,2,22]]},"article-number":"P1.30"}}