On multidimensional covering systems of congruences (Q1071054)

A system of congruences (1) \(\sum^{k}_{j=1}a_{ij} x_ j\equiv 0\) (mod \(m_ i)\), \(1\leq i\leq n\) is said to cover a k-dimensional cube \(C_ k\) if each of its k-dimensional vectors satisfies at least one of the congruences in (1). The author proved earlier [Acta Arith. 43, 191-208 (1984; Zbl 0531.10056)] that if such a system covers a cube \(C_ k\) with side length \(2^{n-1}\) and containing the origin \([0,0,...,0]\) then every k-dimensional integer vector satisfies at least one congruence in (1). The question is what is the minimal length of the side of such a cube sufficient for the same conclusion. Since \(2^{n-2}+1\) is not enough (ibid.), the author conjectured that the answer to this question is \(2^{n-2}+2\). In this paper he proves that if this is not true then there exists a system (1) which does not satisfy the conjecture but with all the moduli under a given bound expressed in terms of k and n.