Arithmetic-progression-weighted subsequence sums

Abstract

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 ≠ wj for i ≠ j}, which is a particular kind of weighted restricted sumset. We show that {pipe}W ⊙ S{pipe} ≥ min{{pipe}G{pipe} - 1, n}, that W ⊙ S = G if n ≥ {pipe}G{pipe} + 1, and also characterize all sequences S of length {pipe}G{pipe} with W ⊙ S ≠ 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 {pipe} n2 and n2 ≥ 3) having k distinct terms, for any k ε [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. © 2012 Hebrew University Magnes Press.

Publication Title

Israel Journal of Mathematics

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