Reconstructing Integer Sets From Their Representation Functions
Reconstructing Integer Sets From Their Representation Functions
复制标题
DOI：
10.37236/1831
复制
发表时间：
2004-11
期刊：
影响因子：
--
通讯作者：
V. Lev
中科院分区：
文献类型：
--
作者：
V. Lev
我们给出了Dombi和Chen和Wang最近关于$a_1+a_2$形式的整数表示的个数的一个简单而普遍的证明，其中$a_1$和$a_2$是给定的无穷整数集的元素。考虑到类似的差集问题，我们证明了正整数集存在一个划分${\bbbN}=\cup_{k=1}^\inty A_k$，使得每个$A_k$都是一个完美差集(这意味着任何非零整数都有唯一的表示为$a_1-a_2$，其中$a_1，a_2\在A_k$中)。提出了一些有待解决的问题。
We give a simple common proof to recent results by Dombi and by Chen and Wang concerning the number of representations of an integer in the form $a_1+a_2$, where $a_1$ and $a_2$ are elements of a given infinite set of integers. Considering the similar problem for differences, we show that there exists a partition ${\Bbb N}=\cup_{k=1}^\infty A_k$ of the set of positive integers such that each $A_k$ is a perfect difference set (meaning that any non-zero integer has a unique representation as $a_1-a_2$ with $a_1,a_2\in A_k$). A number of open problems are presented.