Arithmetic-Progression-Weighted Subsequence Sums

2011/02/25 by Grynkiewicz, David J., Philipp, Andreas, Ponomarenko, Vadim
#11B50 #11B75 #11D79 #11P70 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1102.5351

Abstract

Let G be an abelian group, let S be a sequence of terms s₁,s₂,...,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=\w₁s₁+...+w_ns_n: w_i a term of W, w_i≠ w_jfor i≠ j\, which is a particular kind of weighted restricted sumset. We show that |W\odot S|≥ min\|G|-1, n\, that W\odot S=G if n≥ |G|+1, and also characterize all sequences S of length |G| with W\odot S≠ G. This result then allows us to characterize when a linear equation a₁x₁+...+a_rx_r≡ α\mod n, where α,a₁,..., a_r∈ \Z are given, has a solution (x₁,...,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≅ C_n1⊕ C_n2 (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.

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