On Bialostocki's conjecture for zero-sum sequences

Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\sum_{k=1}^n a_k\equiv\sum_{k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $σ$ on {1,...,n} such that $\sum_{k=1}^n w_k a_{σ(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.

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