NDLI: Arithmetic-progression-weighted subsequence sums

Content Provider	Springer Nature Link
Author	Grynkiewicz, David J. Philipp, Andreas Pomarenko, Vadim
Copyright Year	2012
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.
Starting Page	359
Ending Page	398
Page Count	40
File Format	PDF
ISSN	00212172
Journal	Israel Journal of Mathematics
Volume Number	193
Issue Number	1
e-ISSN	15658511
Language	English
Publisher	The Hebrew University Magnes Press
Publisher Date	2012-09-20
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics Algebra Group Theory and Generalizations Analysis Applications of Mathematics Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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