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\section{The Green--Tao Theorem}

\begin{fcthm}[]
  % Theorem 3.1
  \label{thm:3.1}
  For all $k \ge 3$, the primes contain an arithmetic progression of length $k$.
\end{fcthm}

\begin{fcthm}[Weighted Szemerédi]
  % Theorem 3.2
  \label{thm:3.2}
  For all $k \ge 3$ and $\alpha > 0$, there exists $c = c(k, \alpha) > 0$ such
  that for any function $f : \Zbb / N \Zbb \to [0, 1]$ satisfying $\Ebb f =
  \alpha$,
  \[
    \Ebb_{x, d} f(x) f(x + d) \cdots f(x + (k - 1)d)
    > c(k, \alpha) - o_{k, \alpha}(1)
  ,\]
  where $o_{k, \alpha}(1)$ is a quantity that goes to $0$ as $n \to \infty$ at a
  rate that depending on $k$ and $\alpha$.
\end{fcthm}

There is a standard method that allows converting between a statement like the
above about $\Zbb / N\Zbb$ into a statement about $[N]$:
given a subset of $[N]$, just view it as a subset of $\Zbb / 3N\Zbb$ (arithmetic
progressions in $\Zbb / 3N \Zbb$ that lie in $[N]$ correspond to genuine
arithmetic progressions in $[N]$).

Insight of Green and Tao: the primes are dense in a certain subset of the
naturals.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/b34efef8a2bd4201.png}
\end{center}

We will use some results of Conlon--Fox--Zhao which are sparse versions of the
hypergraph regularity results we saw earlier.
We will only prove one of these results in lectures, which is the sparse
counting lemma.

\begin{fcdefn}[]
  \glsnoundefn{lfc}{$k$-LFC}{}%
  % Definition 3.3
  \label{defn:3.3}
  A function $\nu = \nu^{(N)} : \Zbb / N \Zbb \to [0, \infty)$ is said to
  satisfy the \emph{$k$-linear forms condition ($k$-LFC)} if
  \[
    \Ebb_{\substack{x_1^{(0)}, x_1^{(1)} \\ \vdots \\ x_k^{(0)}, x_k^{(1)}}}
    \prod_{j = 1}^k \prod_{\omega \in \{0, 1\}^{[k] \setminus \{j\}}} \nu \left(
    \sum_{i = 1}^{k} (i - j) x_i^{(\omega_i)} \right)^{n_{j, \omega}}
    = 1 + o(1)
  \]
  for any choice of exponents $n_{j, \omega} \in \{0, 1\}$.
\end{fcdefn}

\begin{example}
  % Example 3.4
  \label{eg:3.4}
  We will omit writing out the $n_{j, \omega}$ in this example.

  $\nu$ satisfies the \lfc{2} if
  \[
    \Ebb_{\substack{x, x' \\ y, y'}} \nu(y) \nu(y') \nu(-x) \nu(x')
    = 1 + o(1)
  .\]
  $\nu$ satisfies the \lfc{3} if
  \[
    \Ebb_{\substack{x, x' \\ y, y' \\ z, z'}}
    \nu(y + 2z) \nu(y' + 2z) \nu(y + 2z') \nu(y' + 2z')
    \nu(-x + z) \nu(-x' + z) \nu(-x + z') \nu(-x' + z')
    \nu(-2x - y) \nu(-2x' - y) \nu(-2x - y') \nu(-2x - y')
    = 1 + o(1)
  .\]
  $\nu(y + 2z) \nu(-x + z) \nu(-2x - y)$ relates to three term progressions: it
  is the arithmetic progression $y + 2z + n(-x - y - z)$ for $n = 0, 1, 2$.
\end{example}

\begin{fcthm}[Relative Szemerédi]
  % Theorem 3.5
  \label{thm:3.5}
  Let $k \ge 3$ and $\alpha > 0$ and suppose $\nu = \nu^{(N)} : \Zbb / N \Zbb
  \to [0, 1]$ satisfies the \lfc{k}.
  Suppose $N$ is sufficiently large and coprime to $(k - 1)!$.
  Let $f : \Zbb / N\Zbb \to [0, \infty)$ satisfies $0 \le f(x) \le \nu(x)$ for
  all $x \in \Zbb / N \Zbb$ and suppose $\Ebb f \ge \alpha$.
  Then
  \[
    \Ebb_{x, d} f(x) f(x + d) \cdots f(x + (k - 1)d)
    \ge c(k, \alpha) - o_{k, \alpha, \ub{\nu}_{\text{rate of convergence in
    \lfc{k}}}}(1)
  \]
  where $c(k, \alpha)$ is as in \cref{thm:3.2}.

  Often refer to $\nu$ as a \emph{pseudorandom majorant} for $f$.
\end{fcthm}

Note: taking $f$ here to be the indicator of the primes won't satisfy the
conditions above, because we need $\Ebb f \ge \alpha$.
Instead, we'll use the fact that $f$ is not bounded (takes values in $[0,
\infty)$).

Think of ``$f(n) = \indicator{\text{primes}}(n) \cdot \log n$.''

In the original Green--Tao proof, they needed an additional condition as well as
the \lfc{k}, and this additional requires a lot of analytic number theory to
prove.
One of the great contributions of the proof by Conlon--Fox--Zhao was to remove
this additional condition, and in particular greatly reducing the amount of
analytic number theory needed to prove the result.

Consider the van Mangoldt function
\[
  \Lambda(n)
  = \begin{cases}
    \log p
    & \text{if $n = p^k$ for some prime $p$} \\
    0
    & \text{otherwise}
  \end{cases}
\]
By the Prime Number Theorem,
\[
  \Ebb_{n \in [N]} \Lambda(n)
  = 1 + o(1)
.\]
(Remark: we won't need the Prime Number Theorem to prove Green--Tao)

\textbf{Problem:} $\Lambda$ is biased with respect to small residue classes.
Use $W$-trick: let $w = w(N)$ be a function $\to \infty$ with $N$ e.g. $\sim \log
\log N$.
Let $W = \prod_{p \le w} p$, and consider only primes $\equiv 1 \pmod{W}$, by
defining
\[
  \tilde{\Lambda(n)}
  = \begin{cases}
    \frac{\varphi(W)}{W} \log(Wn + 1)
    & \text{if $Wn + 1$ is prime} \\
    0
    & \text{otherwise}
  \end{cases}
\]
where $\varphi$ is the Euler totient function.
Can show $\Ebb_{n \in [N]} \tilde{\Lambda}(n) = 1 + o(1)$, provided $w$ grows
sufficiently slowly.