关于Erds-Ginzburg-Ziv定理的一个注记(英文)

A Note on the Erds-Ginzburg-Ziv Theorem

设G为n阶加法Abe1群 ,S ={ai} 2n- 1 i=1 是G中元序列 ,对a∈G用r(S ,a)表示a写成S中n项之和的方法数 .196 1年Erd s ,Ginzburg与Ziv证明了n为素数时r(S ,0 )≥ 1.1996年高维东指出n是素数 p时 r(S ,0 )≡ 1(mod p) .证明了下述结果 :假定有特征为素数 p的域使G为其加法子群 ,则r(S ,0 )≡ 1(modp) ,且对a∈G \{ 0 }有r(S ,a)≡ 0 (mod p) .这推广了高维东的工作 .

Let G be an additive abelian group of order n, and S={a i} 2n-1 i=1 a sequence of 2n-1 elements in G. For a∈G let r(S,a) denote the number of ways we can write a as a sum of n terms of S. If G is a subgroup of the additive group of F, where F is a field of prime characteristic p , then we have r(S,0)≡1 (mod p), and r(S,a)≡0 (mod p) for a∈G\{0}. This extends Gao's work on a theorem of Erds, Ginzburg and Ziv.