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.**