Elon Lindenstrauss explains in his [talk](http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/4043/show_video) 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?



Additive combinatorics and large Fourier coefficients
I think I figured it out myself. What was meant is that for every finite subset $S$ of $A\_{\delta}$ one has
$$| \lbrace (n,m) \in S \times S \mid n-m \in A\_{\delta^2/2} \rbrace | \geq \frac{\delta^2 |S|^2}{2}.$$
This follows from the proof of the second part of Lemma 4.37 in *Tao and Vu, Additive Combinatorics*. (There, the inequality is stated incorrectly as $\leq$.)


