An addition theorem for finite cyclic groups (Q1356553)

The Erdős-Ginzburg-Ziv theorem states that if \(a_1, a_2,\dots, a_{2n-1}\) is a sequence of \(2n-1\) elements in a finite abelian group \(G\) of order \(n\) (written additively), then \(0= a_{i_1}+\cdots+ a_{i_n}\) with \(1\leq i_1<\cdots< i_n\leq 2n-1\). In this paper the author proves the following improvement of this result. Theorem. Let \(n\geq 2\) and \(2\leq k\leq [n/4]+2\), and let \(a_1, a_2,\dots, a_{2n-k}\) be a sequence of \(2n-k\) elements in the cyclic group \(Z_n\) of order \(n\). Suppose that for any \(n\)-subset \(I\) of \(\{1,\dots, 2n-k\}\), \(\sum_{i\in I}a_i\neq 0\). Then, one can rearrange the sequence of the type \[ \undersetbrace u\to{a,\dots, a}, \undersetbrace v\to{b,\dots, b}, c_1,\dots, c_{2n-k-u-v}, \] where \(u\geq n-2k+3\), \(v\geq n-2k+3\), \(u+v\geq 2n-2k+2\) and \(a-b\) generates \(Z_n\).