关于$\mathbb{R}$中康托尔集中点的测度
On the Measure of the Midpoints of the Cantor Set in $\mathbb{R}$
摘要：
本文主要讨论如下问题：设$T$是$\mathbb{R}^n$中的一个固定集.设$S$和$B$是$\mathbb{R}^n$中的两个子集，使得对于$S$中的任何$x$，存在一个$r$，使得$x+ r T$是$B$的子集。如果我们知道了$S$的大小，那么$B$能有多小呢？Stein利用球面极大算子证明了：若$n$大于或等于3，且$T$是以原点为中心的球面，则$S$有正测度蕴涵$B$有正测度.后来，Bourgain和Marstrand证明了$n =2$的类似结果。我们找到了一个例子来解释为什么$n=1$的结果失败了。
英文摘要：
In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $\mathbb{R}^n$. And let $S$ and $B$ he two subsets in $\mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of $B$. How small can be $B$ be if we know the size of $S$? Stein proved that for $n$ is greater than or equal to 3 and $T$ is a sphere centered at origin, then $S$ has positive measure implies $B$ has positive measure using spherical maximal operator. Later, Bourgain and Marstrand proved the similar result for $n =2$. And we found an example for why the result fails for $n=1$.
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