On Bialostocki's conjecture for zero-sum sequences

Abstract

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 $\sigma$ on {1,...,n} such that $\sum_{k=1}^n w_k a_{\sigma(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.Comment: 6 page

Arithmetic-progression-weighted subsequence sums

“…In the notation of the following section, our main result is as follows. It is worth noting that Theorem 1.1 contains, as a very special case, the main result from [31], which was devoted to proving the aforementioned conjecture of Bialostocki in the case when the weight sequence is an arithmetic progression of even difference. Theorem 1.1.…”

“…With this notation, the weighted Erdős-Ginzburg-Ziv Theorem says that if W is any zero-sum modulo |G| sequence of integers and S is a sequence of terms from G with length |S| ≥ 2|G| − 1, then S has a |G|-term subsequence S ′ with 0 ∈ W ⊙ S ′ . It is still an open conjecture of Bialostocki that the weaker hypothesis |S| = |G| with S zero-sum is enough to guarantee 0 ∈ W ⊙ S when |G| is even [10] [31].…”