I wrote a short note entitled Covering the large spectrum and generalized Riesz products that simplifies and generalizes the approach of the first few posts on Chang’s Lemma and Bloom’s variant.

The approximation statement is made in the context of general probability measures on a finite set (though it should extend at least to the compact case with no issues). The algebraic structure only comes into play when the spectral covering statements are deduced (easily) from the general approximation theorem. The proofs are also done in the general setting of finite abelian groups.

Comments are encouraged, especially about references I may have missed.

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A very beautiful and short proof of Chang’s and Bloom’s lemma. I just want to ask why η should satisfy $η<1/e^3$ in Theorem 2.1? Is this condition necessary？

That condition is there just so that log log(1/eta) is well-defined. On the other hand, the theorem is only non-trivial for eta < 1, so this is not really a restriction (since there is already a leading constant factor of 9).

Thanks very much for your clarifications Prof James. I also think so and belive your approximation theorem in this paper have more applications.