Counting points in hypercubes and convolution measure algebras (Q1078567)

Let A and B be non-empty subsets of \(\{0,1\}^ n\), \(\alpha =(1/2)\log_ 2 3\). Then it is shown that the sumset \(A+B\) has cardinality \(| A+B| \geq (| A| | B|)^{\alpha},\) as had been conjectured by Erdős in the case \(A=B\). Necessary and sufficient conditions for equality are also given. The following result is proved as an application: Let \(\lambda\) be Lebesgue measure, and \(\mu\) Haar measure on the Cantor group \(D=\{0,1\}^{{\mathbb{N}}}\), identified with the Cantor set in [0,1]. If \(X,Y\subseteq D\) are analytic sets with \(\mu(X)\mu(Y)>0\) then \(\lambda(X+y)\geq 2\mu(X)^{\alpha}\mu (Y)^{\alpha}.\)