Counting points in hypercubes and convolution measure algebras

It is shown that if A and B are non-empty subsets of {0, 1} ⁿ (for some nεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points of A lie on a hyperplane in n-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: if X and Y are compact subsets of K (the Cantor set) with m(X), m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α where m is the usual measure on K and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing that m is not concentrated on any proper Raikov system.