Arithmetic-Progression-Weighted Subsequence Sums
David J. Grynkiewicz, Andreas Philipp, Vadim Ponomarenko

TL;DR
This paper establishes bounds and characterizations for weighted sumsets in abelian groups, linking these results to solutions of linear equations modulo n and properties of zero-sum sequences.
Contribution
It introduces new bounds and characterizations for weighted sumsets in abelian groups, and applies these to solve linear congruences and analyze zero-sum sequences.
Findings
|W⊙S| ≥ min{|G|-1, n} for certain sequences
W⊙S=G if n ≥ |G|+1
Existence of maximal length minimal zero-sum sequences with arbitrary pattern of multiplicities
Abstract
Let be an abelian group, let be a sequence of terms not all contained in a coset of a proper subgroup of , and let be a sequence of consecutive integers. Let which is a particular kind of weighted restricted sumset. We show that , that if , and also characterize all sequences of length with . This result then allows us to characterize when a linear equation where are given, has a solution modulo with all distinct modulo . 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…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · graph theory and CDMA systems
