The linear diophantine problem of Frobenius for subsets of arithmetic sequences (Q1364581)

Let \(A_k= \{a_1, \dots, a_k\} \subset N\) with gcd \((a_1, \dots, a_k)=1\) and \(g(A_k)\) the corresponding Frobenius number. Now the special case \(A_k= \{a,ha+ d,ha + 2d, \dots, ha+ (k-1)d\}\) with \(h,d>0\) and gcd \((a,d) =1\) is considered, and \(\ell_k\) is defined as the greatest number of elements which can be omitted without altering \(g(A_k)\). Then it is shown that \[ 1-{4\over \sqrt k} \leq{\ell_k\over k} \leq 1-{3 \over k} \] provided \(a>k\), or \(a=k\) with \(d>2h \sqrt k\). In the case \(a> (k-4)k+3\) the lower bound can be improved to \(1-{4 \over k}\). Moreover, sets \(E_k\subset A_k\) are determined such that \(g(A_k \backslash E_k) =g(A_k)\).