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Seminar program - season 2

We are following the Tao-Teräväinen paper: https://arxiv.org/abs/2107.02158 .

Tao's blog post about the above paper might also help: https://terrytao.wordpress.com/2021/07/05/

Table of Contents


Seminar Program - Part I;      I-1,      I-2,      I-3

Part II;     II-1,     II-2,     II-3

Part III


Write $Q:= \exp ( (\log N)^{1/10} )$.

Note that Q defined in this way grows slower than any positive power of $N$ because $N^{c} = \exp (c \log N)$ and $c\log N > (\log N )^{1/10}$. It grows faster than any power of $\log N$ because $\log ^A N := (\log N)^A = \exp (\log ( \log ^A N ) ) = \exp ( (\log \log N ) + \log A ) $ and we know $(\log N )^{1/10 } > \log \log N $.

The main result is the bound of the following Gowers uniformity norm, with some sufficiently small $c>0$:

\[ \| \Lambda - \Lambda _{Cramér,Q}  \| _{U^k ([N])} \ll (\log \log N )^{-c} +Q^{-c} .   \]

Let's write $\| - \| := \|  - \| _{U^k([N])}$ to make it easier to type.

By the triangle inequality of $\| - \| _{U^k}$, this would follow from:

\[ \|   \Lambda - \Lambda _{Siegel} \| \ll (\log \log N)^{-c}  \quad \text{and }\quad \|  \Lambda _{Siegel} - \Lambda _{Cramér,Q} \| \ll Q^{-c} . \]

The latter inequality is easier and proved in §5.


The former requires substantial work.

By Manners' form of GI(s) explained in §6, this will follow from Theorem 2.7:

for certain nilsequences $\mathbb Z \xrightarrow{g} G\to G/\Gamma \xrightarrow{F}\mathbb C$ and arithmetic progressions $P\subset [N] $, we have

\[ \sum _{n\in P} (\Lambda - \Lambda _{Siegel})(n) \overline F (g(n) ) \ll N\exp ( (-\log N)^{0.09 } ) . \]

This estimate is the main part of the paper. The proof requires §5 + §7 + former Green-Tao work on MN(s).


By the way, §1 of the paper is introduction, §2 is an extended introduction including def of $\Lambda _{Siegel}$, §3 is a one-page section about the notation, §4 is recollection of $\| - \| _{U^k([N])}$.

The main content spans through §§5-8 (as we will see below). §9 explains the application to linear equation in primes. The final section §10 explains another application.

Seminar Program - Part I

The following topics can be studied independently, so let's distribute them and give lectures to each other:

Basics of $\Lambda _{Cramér,w}$

Cover §5.1. The computation is annoying. But eventually we'll have to be comfortable with this type of computation.

Basics of $L (s,\chi )$ and Siegel zeros

What is $L(s,\chi )$, to begin with? (Read §5.9 of the Iwaniec-Kowalski book, say.)

What are the (Landau-)Siegel zeros? (Read p.106 of the Iwaniec-Kowalski book, say.)

Theorem 2.7 (p.122) of that book (which should motivate $\Lambda _{Siegel} $).

Wikipedia would help too. https://en.wikipedia.org/wiki/Siegel_zero

Manners' GI(s)

State Theorem 6.2 of Tao-Teräväinen. To do so, you will have to recall relevant notions for the audience. (I believe all the relevant notions can be found within the paper.)

[optional] Explain the implication (Theorem 2.7 of Tao-Teräväinen) $\Rightarrow $ $\| \Lambda - \Lambda _{Siegel} \|_{U^k([N])}  \ll (\log \log N )^{-c} $, perhaps ignoring the "boundedness" issue for now.

[optional but preferred] What would be Manners' GI(s) for $\mathbb Z^n $ if we implement this blog post of Tao? https://terrytao.wordpress.com/2015/07/24/ 


Part II

Having gone through Part I, we will be ready to study the following topics. Again, they can be studied independently of each other.

Toward the conparison of $\Lambda _{Siegel}$ and $\Lambda _{Cramér,Q}$

Present the first half of §5.2 (p. 20 -- first quarter of p. 22).

This seems to depend on what we have learned about Siegel zeros in Part I.

The first 1/3 of §7

Here, they prove Prop. 2.2, the main theorem with the test function $F$ constant.

We want to understand this part in depth.

Manners' GI(s) continued

Read §8, where they modify Manners' GI(s) so that it can be applied to non-bounded functions like $\Lambda - \Lambda _{Siegel}$.

Draw a Leitfaden of §8.

You don't necessarily have to follow everything logically but it would be nice if you could determine what are the key external inputs.


Part III


to be continued