A rather specific question, concerning the second remark of Tao in
As I interpret it, the second remark states that the Theorem which begins the note may alternatively be derived by an application of two other results used as black boxes, although the alternative proof occupying the note has the advantage of being more flexible. I have not been able to construct an argument deducing the Theorem from the results quoted in this second remark (the 'Fourier reduction lemma' and Halasz's theorem): might someone be able to say what I'm missing?
Let me give a little of the background. This note by Tao was written during the Polymath project some five years ago on the Erdos discrepancy problem. The special case -- showing that a completely multiplicative $\{\pm 1\}$-valued function has unbounded partial sums -- is still interesting, and still unknown. The main theorem of the note states that were such a function $g$ to exist it would not correlate with a non-principal Dirichlet character $\chi$, in the sense that $$\sum\limits_{p}\dfrac{\vert 1-g(p)\overline{\chi(p)}\vert}{p}=\infty$$
The 'Fourier reduction lemma' to which he refers is from here: http://michaelnielsen.org/polymath1/index.php?title=Fourier_reduction
-- another ingenious argument of Tao reducing the general Erdos discrepancy problem to a similar statement about completely multiplicative $S^1$-valued functions. 'Theorem 2.3' from Balog, Granville and Soundararajan is just a version of Halasz's Theorem on sums of multiplicative functions.
I've read what I believe to be the corresponding discussion on the original blog post (EDP4 on Gowers' blog), but have not been able to reconstruct an appropriate argument. No doubt this is due to my own innate stupidity.