Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Arithmetic-Progression-Weighted Subsequence Sums   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 $$W\odot
S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$
which is a particular kind of weighted restricted sumset. We show that $|W\odot
S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also
characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This
result then allows us to characterize when a linear equation
$$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ 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
$G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$
distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence.

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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex