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?
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