Sections 4 & 5 of the Tao-Teravainen paper
https://arxiv.org/abs/2107.02158
Recall that one of the goals of this paper is to compute the sum
\[ \sum _{n\in \Omega \cap \mathbf Z ^m} \prod _{i=1}^t \Lambda (\psi _i (n) ) \]
where $\Omega \subset \mathbf R^m $ is a bounded convex set, $\Lambda $ is the von Mangoldt function (which basically detects prime numbers) and $\psi _i \colon \mathbf Z ^m \to \mathbf Z $ are a tuple of $\mathbf Z $-linear maps whose linear parts are pairwise linearly independent over $\mathbf Q $.
Section 4
This 3-page section states three lemmas and corollary on the Gowers uniformity norm. All of them are elementary in nature (but unfortunately not easy). It seems one should check out their statements but we don't have to cover this section in dedicated lectures.
Section 5
In this section they do two things.
The Cramer model is computable
In §5.1, they compute the sum
\[ \sum _{n\in \Omega \cap \mathbf Z ^m} \prod _{i=1}^t \Lambda _{Cramer,Q} (\psi _i (n) ) . \]
Thanks to this computation and the theory of Gowers norms, the goal at the top of this page will be reduced to showing that:
* $\Lambda _{Cramer,Q}$ and $\Lambda _{Siegel}$ are close in the Gowers uniformity norm
and
* $\Lambda _{Siegel}$ and $\Lambda $ are also close.
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\[ \Lambda_{\mathrm{Cramer},Q} \leftrightarrow \Lambda _{\mathrm{Siegel}} \leftrightarrow \Lambda \quad\text{ Are they close in }\Vert - \Vert _{U^{s+1}} ? \] ===========================================================
Task 1 (workload = ***)
Explain the computation of the above sum $\sum _{n\in \Omega \cap \mathbf Z ^m} \prod _{i=1}^t \Lambda _{Cramer,Q} (\psi _i (n) )$.
The Cramer model is close to the Siegel model
In §5.2, they show that $\Lambda _{Cramer,Q}$ and $\Lambda _{Siegel}$ are close (Proof of Theorem 2.5, which begins on p. 22).
Though this subsection lasts until p. 26, we only have to follow it up to p. 23 because the rest of it deals with the Mobius function $\mu $.
The technical key of the proof seems to be the Weil bound for character sums (used to prove Lemma 5.6).
Task 2 (workload = ** or more?)
State the Weil bound for character sums. Use google search or the Iwaniec-Kowalski book (Cor. 11.24) to figure out its statement.
Give some comments on its proof (Is it easy? Does it follow from a famous theorem? Should we give up?) or provide some sample computations.
Task 3 (workload = *)
Prove Lemma 5.6 (p. 21) using the Weil bound.
Lemma 5.6 proves the bound
\[ \Vert \chi \Vert _{U^{k} (\mathbf Z / q \mathbf Z) } \ll _{\varepsilon } q^{-\frac{1}{2^{k+1} } +\varepsilon } \]
for any quadratic character $\chi $ on $\mathbf Z / q \mathbf Z $ and $\varepsilon >0$.
Task 4 (workload = **)
Explain Proof of Theorem 2.5 (pp. 22-23).
As the only difference between $\Lambda _{Cramer, Q} $ and $\Lambda _{Siegel}$ is the contribution from the Siegel character $\chi _{Siegel}$, it's not surprising that Lemma 5.6 can be used to prove that this difference has small Gowers norm.
Caution
Don't think about generalizing this story until you have understood it well enough!
Further reading
Proving that $\Lambda _{Siegel}$ and $\Lambda $ are close requires much more work. By the inverse Gowers theory of Green-Tao-Ziegler and Manners, the problem is reduced to another somewhat concrete estimate, and this is settled in §7 where some crucial computations are borrowed from an earlier paper of Green-Tao.
