Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the torus ${\mathbb T}^n$ respects the additive structure. More precisely, he defines

$$A_{\delta} := \lbrace b \in {\mathbb Z}^n \mid |\hat \mu(b)| \geq \delta \rbrace$$

and says that it is "morally" true that $A_{\delta} - A_{\delta} \subset A_{\delta^2}$. (Here, the difference of two subsets is defined to be the set of all possible differences of elements in the respective subsets.) The precise statement (according to Lindenstrauss) is a consequence of the Balog-Szemeredi-Gowers Lemma.

Can someone provide the precise statement or give some hint how the lemma can be used to obtain bounds on Fourier coefficients?