Arithmetic-progression-weighted subsequence sums

Abstract

Let G be an abelian group, let s be a sequence of terms s 1, s 2, …, s n ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let $$W \odot S = \left\{ {w_1 s_1 + \cdots + w_n s_n :w_i a term of W,w_i \ne w_j for i \ne j} \right\},$$ which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| − 1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S ≠ G. This result then allows us to characterize when a linear equation $$a_1 x_1 + \cdots + a_r x_r \equiv \alpha mod n,$$ where α, a 1, …, a r ∈ ℤ are given, has a solution (x 1, …, x r ) ∈ ℤ r modulo n with all x i distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $$G \cong C_{n_1 } \oplus C_{n_2 }$$ (where n 1 |n 2 and n 2 ≥ 3) having k distinct terms, for any k ε [3, min{n 1 + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.