Eberly College of Arts and Sciences
For any subset C of R there is a subset A of 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 of C with A+A Bernstein in I.
Digital Commons Citation
Ciesielski, Krzysztof, "Measure Zero Sets With Non-measurable Sum" (2001). Faculty & Staff Scholarship. 839.