On the estimates for the number of solutions of an equation with integral coefficients (Q1896937)

Let \(I_{n,k} (a_1, \dots, a_n, b)\) denote the number of solutions of \(a_1 x_1+ \cdots+ a_n x_n =b\), where \(a_i,b\in \mathbb{Z}\), \(x_i\in \{0,1, \dots, k-1\}\), \(i=1, \dots, n\) and \(J_{n,k}= \max_{a_i,b\in \mathbb{Z}} I_{n,k} (a_1, \dots, a_n, b)\). Then it is shown: (i) For \(k=\text{const.}\) and \(n\to \infty\), \(J_{n,k} \sim \sqrt {6/\pi} k^n/ \sqrt {n (k^2-1)}\). (ii) For any natural \(n\geq 1\), \(k\geq 2\), \(J_{n,k}< k^n/ \sqrt {n}\).