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• W2950494798 abstract "Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$Wodot S={w_1s_1+...+w_ns_n:;w_i {a term of} W,, w_ineq w_j{for} ineq j},$$ which is a particular kind of weighted restricted sumset. We show that $|Wodot S|geq min{|G|-1,,n}$, that $Wodot S=G$ if $ngeq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $Wodot Sneq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_requiv alphamod n,$$ where $alpha,a_1,..., a_rin Z$ are given, has a solution $(x_1,...,x_r)in Z^r$ 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 $Gcong C_{n_1}oplus C_{n_2}$ (where $n_1mid n_2$ and $n_2geq 3$) having $k$ distinct terms, for any $kin [3,min{n_1+1,,exp(G)}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence." @default.
• W2950494798 created "2019-06-27" @default.
• W2950494798 creator A5040241644 @default.
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• W2950494798 date "2011-02-25" @default.
• W2950494798 modified "2023-09-27" @default.
• W2950494798 title "Arithmetic-Progression-Weighted Subsequence Sums" @default.
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• W2950494798 hasPublicationYear "2011" @default.
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