Remark on linear forms (Q1924579)

Let \(A=\{a_1, \dots, a_k\} \subset\mathbb{N}\), \(m\subset\mathbb{N}\) such that \(\text{gcd} (ma_1,a_2-a_1\), \(a_3-a_2, \dots, a_k-a_{k-1})=1\). For \(j\in\{1, \dots, m\}\) a given number \(n\in\mathbb{N}_0\) is called \(j\)-omitted if it has no representation (1) \(n=\sum^k_{i=1} x_ia_i\), \(x_i \in\mathbb{N}_0\) such that (2) \(\sum^k_{i=1} x_i\equiv j \bmod m\). A number \(n\in\mathbb{N}_0\) is called omitted if it is \(j\)-omitted for some \(j\in\{1, \dots, m\}\). Formulas for \(\omega (m,A)\), the number of omitted \(n\in\mathbb{N}_0\), and \(g(m,A)\), the greatest omitted number, are given in the case that \(A\) is an arithmetic progression. In this case there is just one \(j\) \((1\leq j\leq m)\) such that there is no representation (1) with (2) for \(n=g(m,A)\), and this \(j\) is given explicitly.