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arXiv 1102.5351math.NT
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等差数列加权子序列和

Arithmetic-Progression-Weighted Subsequence Sums

David J. Grynkiewicz, Andreas Philipp, Vadim Ponomarenko

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摘要：

设$G$是一个阿贝尔群，$S$是一个不全包含在$G$的一个真子群的陪集中的项序列$s_1,s_2,...,s_{n}\in G$，$W$是一个$n$连续整数序列。设$$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$是一类特殊的加权限制和集.我们表明，$|W\odot S|\geq \min\{|G|-1,\,n\}$，$W\odot S=G$，如果$n\geq |G|+1$，也具有$W\odot S\neq G$的长度$|G|$的所有序列$S$的特征。然后，这个结果允许我们表征当给出$α,a_1,..., a_r\in \Z$的线性方程$$a_1x_1+...+a_rx_r\equiv α\mod n,$$具有解$(x_1,...,x_r)\in \Z^r$模$n$且所有$x_i$模$n$不同时。作为第二个简单的推论，我们还表明，有最大长度最小零和序列的秩2有限阿贝尔群$G\cong C_{n_1}\oplus C_{n_2}$（其中$n_1\mid n_2$和$n_2\geq 3$）具有$k$不同的条款，任何$k\in [3,\min\{n_1+1,\,\exp(G)\}]$。事实上，除了一些简单的限制，任何模式的多重性是可实现的，这样一个最大长度的最小零和序列。

英文摘要：

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 $$a_1x_1+...+a_rx_r\equiv α\mod n,$$ where $α,a_1,..., a_r\in \Z$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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