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

## TL;DR

This paper investigates the measure-theoretic properties of midpoints in the Cantor set within real numbers, exploring how the size of certain sets constrains the measure of related sets, extending known results in higher dimensions.

## Contribution

The paper extends measure results related to midpoint sets of the Cantor set in , providing new insights and examples for the case when n=1, where previous results do not hold.

## Key findings

- Stein's result for n  and spherical sets
- Extension of measure results to n=2 by Bourgain and Marstrand
- Counterexample demonstrating failure of the result for n=1

## 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 $x+ r T$ 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 $n=1$.