Bounds on generalized Frobenius numbers (Q627922)

Let \(N\geq 2\) and let \(a_1,\dots,a_N\) be \(N\) positive integers with \(a_i\geq 2\) and such that \(\gcd(a_1,\dots,a_N)=1\). The Frobenius number \(g(a_1,\dots,a_N)\) of this \(N\)-tuple is defined to be the largest positive integer that has no representation as \[ \sum_{i=1}^{N}a_ix_i \] where \(x_1,\dots,x_N\) are nonnegative integers. A comprehensive survey on this problem can be found in [\textit{J. L. Ramírez Alfonsín}, The Diophantine Frobenius problem. Oxford: Oxford University Press (2005; Zbl 1134.11012)]. In this paper, the authors consider a generalization of the Frobenius number that has been introduced by \textit{M. Beck} and \textit{S. Robins} in [``A formula related to the Frobenius problem in two dimensions'', Number theory: New York seminar 2003. New York, NY: Springer, 17--23 (2004; Zbl 1053.11023)]. Let \(s\) be a nonnegative integer. The \(s\)-Frobenius number \(g_s(a_1,\dots,a_N)\) is defined to be the largest positive integer that has precisely \(s\) distinct representations. The main goal in this note is to extend the geometric method of [\textit{L. Fukshansky} and \textit{S. Robins}, Discrete Comput. Geom. 37, No. 3, 471--483 (2007; Zbl 1136.11307)] to obtain general upper and lower bounds on \(g_s\).