Rainbow solutions of linear equations over ℤ_p

We prove that if the group ℤ_p, with p a prime, is coloured with k ≥ 4 different colours such that each colour appears at least k times, then for any a_1, . . . a_k, b in ℤ_p with not all the a_i being equal, we may solve the equation A_(1)x_(1) + • • • + a_(k)x_(k) = b so that each of the variables is chosen in a different colour class. This generalises a similar result concerning three colour classes due to Jungić, Licht, Mahdian, Nešetřil and Radoičić. In the course of our proof we classify, with some size caveats, the sets in ℤ_p which satisfy the inequality | A_1 + • • • + A_n | ≤ | A_1 | + • • • + | A_1 |. This is a generalisation of an inverse theorem due to Hamidoune and Rødseth concerning the case n = 2.