  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 $n=1$.
