A more general result than what you want appears as Theorem 1 in http://arxiv.org/pdf/1002.2554. (A slightly weaker result had appeared before as Theorem 3.1 in http://arxiv.org/pdf/0909.5471v1).

Curiously, it is open if one of $A^2+A^2$ or $A^3+A^3$ is necessarily large.

A more general result than what you want appears as Theorem 1 in http://arxiv.org/pdf/1002.2554. (A slightly weaker result had appeared before as Theorem 3.1 in http://arxiv.org/pdf/0909.5471).

Curiously, it is open if at least one of $A^2+A^2$ and $A^3+A^3$ is necessarily large.