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 λHt(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 λHt(m, n; s, k) in each of the following cases: (i) s, k ≡ 0 (mod 4); (ii) s ≡ 2 (mod 4) and k ≡ 0 (mod 4); (iii) s ≡ 0 (mod 4) and k ≡ 2 (mod 4); (iv) s, k ≡ 2 (mod 4) and m, n both even.