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?
Best Answer
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 $\leq$.)