Measure Zero Sets with Non-Measurable Sum

Abstract

For any $C\subseteq \mathbb{R}$ there is a subset $A\subseteq C$ such that $A+A$ has inner measure zero and outer measure the same as $C+C$. Also, there is a subset $A$ of the Cantor middle third set such that $A+A$ is Bernstein in $[0,2]$. On the other hand there is a perfect set $C$ such that $C+C$ is an interval $I$ and there is no subset $A\subseteq C$ with $A+A$ Bernstein in $I$.