Arithmetic-progression-weighted Subsequence Sums

Let G be an abelian group, let S be a sequence of terms s1, s2, . . . , sn ∈ 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 ̄ S = {w1s1 + . . . + wnsn : wi a term of W, wi 6= wj for i 6= j}, 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 6= G. This result then allows us to characterize when a linear equation a1x1 + . . . + arxr ≡ α mod n, where α, a1, . . . , ar ∈ Z are given, has a solution (x1, . . . , xr) ∈ Zr modulo n with all xi 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 ∼= Cn1 ⊕ Cn2 (where n1 | n2 and n2 ≥ 3) having k distinct terms, for any k ∈ [3, min{n1 + 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.