Associate bases in the postage stamp problem
Let \(h,k\in\mathbb{N}\), \(A_ k=\{a_ 1,a_ 2,\dots,a_ k\}\subseteq\mathbb{N}\) with \(1=a_ 1<a_ 2<\cdots<a_ k\) and \(hA_ k=\left\{\sum^ k_{i=1}x_ ia_ i\;|\;x_ i\in\mathbb{N},\;\sum^ k_{i=1}x_ i\leq h\right\}\). The integer \(n_ h(A_ k)=\max\{n\in\mathbb{N}\;|\;[1,n]\cap\mathbb{N}\subseteq A_ k\}\) is denoted as the \(h\)-range of \(A_ k\). For lower bounds of the extremal \(h\)-range \(n_ k(k)=\max\{n_ k(A_ k)\;|\) \(A_ k\subseteq\mathbb{N},| A_ k|=k\}\) as \(h\to\infty\) special sequences of bases \(A_ k(h)\) (parameter bases) are considered. For \(k\geq 3\) and \(A_ k(h)\) a parameter basis the author constructs another parameter basis \(\hat A_ k(h)\) such that \(n_ h(\hat A_ k(h))\sim n_ h(A_ k(h))\) as \(h\to\infty\).
scientific article; zbMATH DE number 98395
scientific article