Davenport constant with weights

Let A ⊂ { 1 , … , n − 1 } and let ( x 1 , … , x t ) ∈ Z t be a sequence of integers with a maximal length such that for all ( a 1 , … , a t ) ∈ A t , ∑ 1 ≤ i ≤ t a i x i ≢ 0 ( mod n ) . The authors show that for any sequence of integers ( x 1 , … , x n + t ) ∈ Z n + t , there are b 1 , … , b n ∈ A and 1 ≤ k 1 < k 2 < ⋯ < k n ≤ n + t such that ∑ 1 ≤ i ≤ n b i x k i ≡ 0 ( mod n ) .