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.
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Last modified September 13, 2002.