Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays

Let m, n, s, k be integers such that 4≤s≤n,4≤k≤m and ms=nk. Let λ be a divisor of 2ms and let t be a divisor of 2ms/λ. In this paper we construct magic rectangles MR(m, n;s, k), signed magic arrays SMA(m,n;s, k) and integer λ-fold relative Heffter arrays λH_t(m, n;s, k) where s, k are even integers. In particular, we prove that there exists an SMA(m, n;s, k) for all m, n, s, k satisfying the previous hypotheses. Furthermore, we prove that there exist an MR(m, n;s, k) and an integer λH_t(m, n;s,k) in each of the following cases: (i) s, k≡0 (mod4); (ii) s≡2 (mod4) and k≡0 (mod4); (iii) s≡0 (mod4) and k≡2 (mod 4); (iv)s, k≡2 (mod4) and m, n both even.