On the Measure of the Midpoints of the Cantor Set in $\mathbb{R}$

Abstract

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 $x+rT$ 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 $n=1$.

8 pages, 2 figures