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% 24/02/2026 10AM

Reminder: instead of $\Lambda$, want to consider
\[
  \tilde{\Lambda}(n)
  = \begin{cases}
    \frac{\varphi(W)}{W} \log(Wn + 1)
    & \text{if $Wn + 1$ prime} \\
    0
    & \text{otherwise}
  \end{cases}
\]

\begin{fcprop}[Pseudorandom majorant for the primes]
  % Proposition 3.5
  \label{prop:3.5}
  For all $k \ge 3$, there exists $\delta > 0$ such that for all sufficiently
  large $N$, there exists $\nu^{(N)} : \Zbb / N\Zbb \to [0, \infty)$ satisfying
  the \lfc{k} and $\nu(n) \ge \delta_k \tilde{\Lambda}(n)$ for all $n \in [N / 2,
  N)$.
  $\nu$ is given by
  \[
    \nu(n)
    = \begin{cases}
      \frac{\varphi(W)}{W} \frac{\Lambda_{\chi, R}(Wn + 1)^2}{c_\chi \log R}
      & n \in [N / 2, N) \\
      1
      & \text{otherwise}
    \end{cases}
  \]
  where
  \begin{itemize}
    \item $\chi : \Rbb \to [0, 1]$ is a smooth function supported on $[-1, 1]$
      with $\chi(0) = 1$
      \begin{center}
        \includegraphics[width=0.3\linewidth]{images/bb0aa24f94a3436d.png}
      \end{center}
    \item $c_\chi = \int_0^\infty |\chi'(x)|^2 \dd x$
    \item $R = N^{k^{-1} 2^{-k - 3}}$.
    \item $\Lambda_{\chi, R}(n) = \log R \sum_{d \mid n} \mu(d) \chi \left(
      \frac{\log d}{\log R} \right)$.
      \\
      Compare with $\Lambda(n) = \sum_{d \mid n} \mu(d) \log \left( \frac{n}{d}
      \right)$ and $\sum_{\substack{d \mid n \\ d \le R}} \log \frac{R}{d}$.
  \end{itemize}
\end{fcprop}

\begin{proof}
  Omitted.
\end{proof}

Next goal: ``Approximate'' $0 \le f \le \nu$ by $0 \le \tilde{f} \le 1$ with $\Ebb
f = \Ebb \tilde{f}$ (\emph{transference principle}).

\begin{fcdefn}[]
  \glsnoundefn{dp}{DP}{}%
  \glsnoundefn{discp}{discrepancy pair}{}%
  % Definition 3.6
  \label{defn:3.6}
  Let $G$ be an abelian group (which you can think of as being $\Zbb / N\Zbb$),
  let $r \in \Nbb$, and let $\psi : G^r \to G$ a surjective homomorphism, $f,
  \tilde{f} : G \to [0, \infty)$.
  We say $(f, \tilde{f})$ is an \emph{$(r, \eps)$-discrepancy pair (DP) with
  respect to $\psi$} if
  \[
    |\Ebb_{x = (x_1, \ldots, x_r) \in G^r} (f(\psi(x)) - \tilde{f}(\psi(x)))
    \prod_{i = 1}^r u_i(\ub{x_{[r] \setminus \{i\}}}_{= (x_1, \ldots, x_{i - 1},
    x_{i + 1}, \ldots, x_r)})|
    \le \eps
  \]
  for all functions $u_1, \ldots, u_r : G^{r - 1} \to [0, 1]$.
\end{fcdefn}

In words: no function in fewer variables can distinguish $f$ from $\tilde{f}$.

Can think of $\psi(x) = x_1 + x_2 + \cdots + x_r$.

\begin{fcthm}[Dense Model Theorem]
  % Theorem 3.7
  \label{thm:3.7}
  For all $\eps > 0$, there exists $k = \eps^{-O(1)}$ and $\eps' =
  \exp(-\eps^{-O(1)})$ such that the following holds:

  Let $X$ be a finite set, let $\mathcal{F}$ be a collection of functions
  $\varphi : X \to [-1, 1]$.

  Suppose $\nu : X \to [0, \infty)$ satisfying
  \[
    |\langle \nu - 1, \varphi \rangle|
    \le \eps'
    \qquad \forall \varphi \in \mathcal{F}^k = \left\{ \prod_{i = 1}^{k}
    \varphi_i : \varphi_i \in \mathcal{F}, k' \le k \right\}
  \]
  and $f : X \to [0, \infty)$ satisfies $f \le \nu$ and $\Ebb f \le 1$.

  Then there exists $\tilde{f} : X \to [0, 1]$ such that $\Ebb \tilde{f} = \Ebb
  f$ and $|\langle f - \tilde{f}, \varphi \rangle| \le \eps$ for all $\varphi
  \in \mathcal{F}$.
\end{fcthm}

\begin{fccoro}[]
  % Corollary 3.8
  \label{coro:3.8}
  For all $\eps > 0$, there exists $\eps' = \exp(-\eps^{-O(1)})$ such that the
  following holds:

  Let $G$ be an abelian group, $r \in \Nbb$, $\psi : G^r \to G$ a surjective
  homomorphism.
  Let $f, \nu : G \to [0, \infty)$ be such that $0 \le f \le \nu$, $\Ebb f \le
  1$ and $(\nu, 1)$ is an $(r, \eps')$-\gls{discp} with respect to $\psi$.
  Then there exists $\tilde{f} : G \to [0, 1]$ such that $\Ebb \tilde{f} = \Ebb
  f$ and $(f, \tilde{f})$ is an $(r, \eps)$-\gls{discp} with respect to $\psi$.
\end{fccoro}

\begin{proof}[Deduction from \cref{thm:3.7}]
  For any collection of functions $u_1, \ldots, u_r : G^{r - 1} \to [0, 1]$,
  define a \emph{generalized convolution with respect to $\psi$}
  \begin{align*}
    (u_1, \ldots, u_r)_\psi^* : G
    &\to [0, 1] \\
    (u_1, \ldots, u_r)_\psi^* (x)
    &= \Ebb_{\substack{y \in G^r \\ \psi(y) = x}} \prod_{i = 1}^{r} u_i(y_{[r]
    \setminus \{i\}})
  \end{align*}
  $r = 2$:
  \[
    u_1 * u_2(x)
    = \Ebb_{\substack{y \in G^2 \\ y_1 + y_2 = x}} u_1(y_1) u_2(y_2)
  \]
  So indeed the earlier expression looks like a reasonable definition for a
  generalised convolution.

  Notice that the LHS in \cref{defn:3.6} is just $\langle f - \tilde{f}, (u_1
  \cdots u_r)_\psi^* \rangle$.

  Let $\mathcal{F}$ be the set of functions which can be written as convex
  combinations of generalised convolutions with respect to $\psi$.
  Then by the hypotheses, $(\nu, 1)$ is an $(r, \eps')$-\gls{discp} with respect
  to $\psi$, which is equivalent to $\langle \nu - 1, \varphi \rangle \le \eps'$
  for all $\varphi \in \mathcal{F}$.

  Want: $|\langle f - \tilde{f}, \varphi \rangle| \le \eps$ for all $\varphi \in
  \mathcal{F}$.

  So it suffices to show that $\mathcal{F}$ is closed under multiplication.
  Indeed,
  \begin{align*}
    (u_1, \ldots, u_r)_\psi^* (x) (u_1', \ldots, u_r')_\psi^*(x)
    &= \Ebb_{\substack{y \in G^r \\ \psi(y) = x}}
    \Ebb_{\substack{y' \in G^r \\ \psi(y) = x}}
    \prod_{i = 1}^{r} u_i(y_{[r] \setminus \{i\}})
    \prod_{i = 1}^{r} u_i'(\ub{y_{[r] \setminus \{i\}}'}_{= y_{[r] \setminus
    \{i\}} + z_{[r] \setminus \{i\}}}) \\
    &= \Ebb_{\substack{z \in G^r \\ \psi(z) = 0}} \left[
    \ub{\Ebb_{\substack{y \in G^r \\ \psi(y) = x}}
    \prod_{i = 1}^{r} u_i(y_{[r] \setminus \{i\}}) u_i''(y_{[r] \setminus
    \{i\}})}_{\text{generalised convolution wrt $\psi$}}
    \right]
    \qedhere
  \end{align*}
\end{proof}