Weighted sums in finite cyclic groups

In this paper we prove the following result. Let p be a prime, and k a positive integer. Let {w1,…,wk} be a sequence of k integers such that w1+⋯+wk≡0 (mod p2). Then, for every sequence a1,…,ap2+k−1 of p2+k−1 elements in Cp2, there are k distinct indices i1,i2,…,ik such that w1ai1+⋯+wkaik=0. This confirms partly a conjecture by Caro.