关于避免有理距离的集合的一个注记
A note on sets avoiding rational distances
摘要：
本文将给出Ashutosh Kumar最初得到的结果的一个简短证明：对于每个$A\subset \mathbb{R}$，存在充满$A$的$B\subset A$，使得从$B$开始的两个不同点之间的距离都不是有理数.我们将构造$\mathbb{R}$的Bernstein子集，它也避免了有理距离。我们将给出某些情况，其中前一个结果可以推广到$\mathbb{R}^2$的子集，即。e.对于平面的可测子集仍然成立，如果$non(\mathcal{N})=cof(\mathcal{N})$，则对于给定的正外部测度集，我们可以找到它的全子集，这是部分双射并避免了有理距离。
英文摘要：
In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational. We will construct a Bernstein subset of $\mathbb{R}$ which also avoids rational distances. We will show some cases in which the former result may be extended to subsets of $\mathbb{R}^2$, i. e. it remains true for measurable subsets of the plane and if $non(\mathcal{N})=cof(\mathcal{N})$ then for a given set of positive outer measure we may find its full subset which is a partial bijection and avoids rational distances.
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。
