A Weighted Generalization of Gao's n + D − 1 Theorem

Let G denote a finite abelian group of order n and Davenport constant D, and put m = n + D − 1. Let x = (x1,. . .,xm) ∈ Gm. Gao's theorem states that there is a reordering (xj1, . . ., xjm) of x such that $$ \sum_{ 1\leq i\leq n}x_{j_{i}}=0. $$ Let w = (x1, . . ., wm) ∈ ℤm. As a corollary of the main result, we show that there are reorderings (xj1, . . ., xjm) of x and (wk1, . . ., wkm) of w, such that $$ \sum_{ 1\leq i\leq n}w_{k_{i}}x_{j_{i}}=\biggl(\sum_{ 1\leq i\leq n}w_{k_{i}}\biggr)x_{j_{1}}, $$ where xj1 is the most repeated value in x. For w = (1, . . ., 1), this result reduces to Gao's theorem.