Base \(N\) just touching covering systems (Q2754953)

Let \(N\) be a natural number \(> 1\) and let \(A = \{a_1,a_2,\ldots,a_N\}\) be a set of integers such that \(a_i \equiv i \pmod{N}\) for \(i = 1,2,\ldots,N\). The author proves the following theorem: An integer \(m\) is expressible in the form \(m = \sum_{i=0}^k b_i N^i\) for some \(k \geq 0\) with \(b_0,b_1,\ldots,b_k \in \{a - a' \mid a, a' \in A\}\) if and only if it is divisible by the greatest common divisor \(\gcd(a_1 - a_N, a_2 - a_N, \ldots,~{a_N - a_N)}\). So if \(a_N = 0\) and \(\gcd(a_1,a_2,\ldots,a_{N-1}) = 1\) then every integer \(m\) is expressible in the form described above. In this case \(\{a_1,a_2,\ldots,a_N\}\) is a so-called just touching covering system, see \textit{I.\ Kátai} [Generalized Number Systems and Fractal Geometry, Pécs (1995)] and Theorem 6 in \textit{K.-H.\ Indlekofer, I. Kátai, P. Racskó} [Probability Theory and its Applications, Math. Appl. 80, 319-334 (1992; Zbl 0784.11048)]. NEWLINENEWLINENEWLINEThe proof in this paper for general \(N\) is significantly shorter than the proof given by the author for the special case \(N = 3\) [Publ. Math. 51, 241-263 (1997; Zbl 0906.11009)].