论文标题
$ \ mathbb z $的封面与子集总和之间的连接
Connections between covers of $\mathbb Z$ and subset sums
论文作者
论文摘要
在本文中，我们通过残留类别的$ \ Mathbb z $的封面建立了连接，并且在一个字段中的子集总和。假设$ a_0 = \ {a_s（n_s）\} _ {s = 0}^k $覆盖每个整数至少$ p $ time，残留类$ a_0（n_0）= a_0+n_0+n_0 \ n_0 \ mthbb z $ nofredention令$ m_1，\ ldots，m_k \ in \ mathbb z $分别为$ n_1，\ ldots，n_k $。对于任何$ c，c_1，\ ldots，c_k \ in \ mathbb z/p \ mathbb z $ with $ c_1 \ cdots c_k \ c_k \ not = 0 $，我们表明set $ \ bigg \ bigg \ {\ bigg { i} \ frac {m_s} {n_s} \ bigG \}：\，i \ subseteq \ {1，\ ldots，\ ldots，k \} \ \ \ \ mbox {and} \ sum_ sum_} $ 1/n_0 $，其中$ \ {x \} $表示实际数字$ x $的分数部分。
In this paper we establish connections between covers of $\mathbb Z$ by residue classes and subset sums in a field. Suppose that $A_0=\{a_s(n_s)\}_{s=0}^k$ covers each integer at least $p$ times with the residue class $a_0(n_0)=a_0+n_0\mathbb Z$ irredundant, where $p$ is a prime not dividing any of $n_1,\ldots,n_k$. Let $m_1,\ldots,m_k\in\mathbb Z$ be relatively prime to $n_1,\ldots,n_k$ respectively. For any $c,c_1,\ldots,c_k\in\mathbb Z/p\mathbb Z$ with $c_1\cdots c_k\not=0$, we show that the set $$\bigg\{\bigg\{\sum_{s\in I}\frac{m_s}{n_s}\bigg\}:\, I\subseteq\{1,\ldots,k\} \ \mbox{and}\ \sum_{s\in I}c_s=c\bigg\}$$ contains an arithmetic progression of length $n_0$ with common difference $1/n_0$, where $\{x\}$ denotes the fractional part of a real number $x$.
