On exceptional systems of random integers (Q1901506)

The paper deals with the following problem: Let \(n_1, n_2, \dots, n_k\) be positive integers and let \(X_1, X_2, \dots, X_k\) be stochastically independent random integers such that \(X_j\) is uniformly distributed on the set \(\{1, \dots, n_j\}\) for \(j= 1, \dots, k\). A system \((n_1, n_2, \dots, n_k, m)\) is said to be \(u\)-good iff \[ P \Biggl( \sum^k_{j=1} X_j \equiv u\bmod m \Biggr)= {1\over m} \qquad \text{for} \quad 0\leq u\leq m-1. \] Some characterizations of such systems to be \(u\)-good are given. Furthermore, several cases are discussed.