{"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":1753893826336,"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 $p$ and a vector $\\bar\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in {\\Bbb Z}_p^k$ let $f\\left(\\bar\\alpha,p\\right)$ be the largest $n$ such that in each set $A\\subseteq{\\Bbb Z}_{p}$ of $n$ elements one can find $x$ which has a unique representation in the form $x=\\alpha_{1}a_1+\\dots +\\alpha_{k}a_k, a_i\\in A$. Hilliker and Straus bounded $f\\left(\\bar\\alpha,p\\right)$ from below by an expression which contained the $L_1$-norm of $\\bar\\alpha$ and asked if there exists a positive constant $c\\left(k\\right)$ so that $f\\left(\\bar\\alpha,p\\right)&gt;c\\left(k\\right)\\log p$. In this note we answer their question in the affirmative and show that, for large $k$, one can take $c(k)=O(1\/k\\log (2k)) $. We also give a lower bound for the size of a set $A\\subseteq {\\Bbb Z}_{p}$ such that every element of $A+A$ has at least $K$ representations in the form $a+a'$, $a, a'\\in A$.<\/jats:p>","DOI":"10.37236\/1024","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:48:07Z","timestamp":1578718087000},"source":"Crossref","is-referenced-by-count":1,"title":["A Note on a Problem of Hilliker and Straus"],"prefix":"10.37236","volume":"14","author":[{"given":"Miros\u0142awa","family":"Ja\u0144czak","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,10,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1n23\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1n23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:00:57Z","timestamp":1579320057000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1n23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,10,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1024","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,10,30]]},"article-number":"N23"}}