{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:43Z","timestamp":1753893823911,"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>Let $A$ be a subset of the set of nonnegative integers $\\mathbb{N}\\cup\\{0\\}$, and let $r_A(n)$ be the number of representations of $n\\geq 0$ by the sum $a+b$ with $a,b \\in A$. Then $\\big(\\sum_{a \\in A}x^a\\big)^2=\\sum_{n=0}^{\\infty} r_A(n)x^n$. We show that an old result of Erd\u0151s asserting that there is a basis $A$ of $\\mathbb{N}\\cup \\{0\\}$, i.e., $r_A(n) \\geq 1$ for $n \\geq 0$, whose representation function $r_A(n)$ satisfies\u00a0 $r_A(n) &lt; (2e+\\epsilon)\\log n$ for each sufficiently large integer $n$. Towards a polynomial version of the Erd\u0151s-Tur\u00e1n conjecture we prove that for each $\\epsilon&gt;0$ and each sufficiently large integer $n$ there is a set $A \\subseteq \\{0,1,\\dots,n\\}$ such that the square of the corresponding Newman polynomial $f(x):=\\sum_{a \\in A} x^a$ of degree $n$ has all of its $2n+1$ coefficients in the interval $[1, (1+\\epsilon)(4\/\\pi)(\\log n)^2]$. Finally, it is shown that the correct order of growth for $H(f^2)$ of those reciprocal Newman polynomials $f$ of degree $n$ whose squares $f^2$ have all their $2n+1$ coefficients positive is $\\sqrt{n}$. More precisely, if the Newman polynomial $f(x)=\\sum_{a \\in A} x^a$ of degree $n$ is reciprocal, i.e., $A=n-A$, then $A+A=\\{0,1,\\dots,2n\\}$ implies that the coefficient for $x^n$ in $f(x)^2$ is at least $2\\sqrt{n}-3$. In the opposite direction, we explicitly construct a reciprocal Newman polynomial $f(x)$ of degree $n$ such that the coefficients of its square $f(x)^2$ all belong to the interval $[1, 2\\sqrt{2n}+4]$.<\/jats:p>","DOI":"10.37236\/13","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:34:17Z","timestamp":1578713657000},"source":"Crossref","is-referenced-by-count":7,"title":["A Basis of Finite and Infinite Sets with Small Representation Function"],"prefix":"10.37236","volume":"19","author":[{"given":"Art\u016bras","family":"Dubickas","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,1,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:57:37Z","timestamp":1579301857000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/13","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,1,6]]},"article-number":"P6"}}