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% 05/02/2026 10AM

  Want to show:
  \[
    \langle uf, \phi \rangle
    \approx \langle F, \theta \rangle
    = \theta \ub{\langle F, 1 \rangle}_{\le \eta^8 (\delta_{XY} \delta_{XZ}
    \delta_{YZ})^4}
  .\]
  
  \begin{claim}
    % Claim 2.8
    \label{claim:2.8}
    Suppose $2^{38} \eps^{\quarter} \le \sigma \theta$.
    Then $\phi(x, x', y, y', z, z')$ differs from $\theta$ by more than $\sigma
    \theta$ for at most a $\sigma \theta$-proportion of $(x, x', y, y', z, z')
    \in X^2 \times Y^2 \times Z^2$.
  \end{claim}

  Proof: See Example Sheet 1.

  Let $\sigma = \eta^8 (\delta_{XY} \delta_{XZ} \delta_{YZ})^4$.
  Call $(x, x', y, y', z, z')$ \emph{bad} if $|\phi(x, x', y, y', z, z') -
  \theta| > \sigma \theta$, and \emph{good} otherwise.
  Let
  \[
    \tilde{\phi}(x, x', y, y', z, z')
    = \begin{cases}
      \phi(x, x', y, y', z, z')
      & \text{if $(x, x', y, y', z, z')$ is good} \\
      \theta
      & \text{otherwise}
    \end{cases}
  \]
  By the claim, $\|\phi - \tilde{\phi}\|_1 \le 2\sigma \theta$ and
  $\|\tilde{\phi} - \sigma\|_\infty \le \sigma \theta$, and also $\|F\|_1 \le
  1$, $\|F\|_\infty \le 1$.
  Hence
  \begin{align*}
    |\langle F, \phi \rangle - \langle F, \theta \rangle|
    &\le |\langle F, \phi - \tilde{\phi} \rangle|
    + |\langle F, \tilde{\phi} - \theta \rangle| \\
    &\le \|F\|_\infty \|\phi - \tilde{\phi}\|_1
    + \|F\|_1 \|\tilde{\phi} - \theta\|_\infty \\
    &\le 3 \sigma \theta \\
    &= 3 \eta^8 \cdot \delta^8
  \end{align*}
  But
  \[
    |\langle F, \theta \rangle|
    \le \theta \eta^8 (\delta_{XY} \delta_{XZ} \delta_{YZ})^4
    = \eta^8 \cdot \delta^8
  \]
  so
  \[
    |\Ebb_{x, y, z, w} f(x, y, z) g(x, y, w) h(x, z, w) k(y, z, w)|^8
    \le |\langle F, \phi \rangle|
    \le |\langle F, \phi \rangle| + 3\eta^8 \delta^8
    \le 4(\eta \delta)^8
    .
    \qedhere
  \]
\end{proof}

\begin{fccoro}[Simplex counting lemma]
  % Corollary 2.9
  \label{coro:2.9}
  Let $G$ be a $4$-partite graph on $X_1 \cup X_2 \cup X_3 \cup X_4$, with all
  six bipartite graphs \gls{epsquasi} of densities $\delta_{ij}$ respectively.
  Let $H_{ijk}$ be a $3$-partite $3$-uniform hypergraph on $X_i \cup X_j \cup
  X_k$ which is \hquasirel{\eta} $\triangles(G(X_i, X_j, X_k))$, of relative
  density $\delta_{ijk}$.
  Let $H = H_{123} \cup H_{134} \cup H_{124} \cup H_{234}$.
  Then
  \[
    \left|\Ebb_{x_1, x_2, x_3, x_4} \indicator{H}(x_1, x_2, x_3) \indicator{H}(x_1,
    x_2, x_4) \indicator{H}(x_1, x_3, x_4) \indicator{H}(x_2, x_3, x_4)
    - \prod \delta_{ijk} \prod \delta_{ij}\right|
    \le 8 \eta \prod \delta_{ij}
  .\]
\end{fccoro}

\begin{proof}
  Let
  \[
    f(x_1, x_2, x_3)
    = \indicator{H}(x_1, x_2, x_3) - \delta_{123} \indicator{G}(x_1, x_2)
    \indicator{G}(x_1, x_3) \indicator{G}(x_2, x_3)t
  \]
  and consider
  \[
    \Ebb_{x_1, x_2, x_3, x_4} f(x_1, x_2, x_3) \indicator{H}(x_1, x_2, x_4)
    \indicator{H}(x_1, x_3, x_4) \indicator{H}(x_2, x_3, x_4)
  .\]
  By \cref{prop:2.7}, this is at most $2\eta \prod \delta_{ij}$ (in absolute
  value).
  Iterate.
\end{proof}

\begin{fcdefn}[Relative msd]
  \glsadjdefn{relmsd}{mean square density relative to}{}%
  % Definition 2.10
  \label{defn:2.10}
  Let $U$ be a set, let $f : U \to \Rbb$, and let $U = B_1 \cup B_2 \cup \cdots
  \cup B_r$ be a partition of $U$.
  Then the \emph{mean square density (msd)} of $f$ relative to this partition is
  \[
    \sum_{i = 1}^{r}  \frac{|B_i|}{|U|} |\Ebb_{x \in B_i} f(x)|^2
  .\]
\end{fcdefn}

\begin{fclemma}[]
  % Lemma 2.12
  \label{lemma:2.12}
  Let $U$ be a set, $f, g : U \to [-1, 1]$, $U = B_1 \cup B_2 \cup \cdots \cup
  B_r$.
  Suppose $g$ is constant on each $B_i$.
  Then the \relmsd{f} the partition is $\ge \frac{\langle f, g
  \rangle^2}{\|g\|_2^2}$.
\end{fclemma}

\begin{fcdefn}[]
  \glsnoundefn{index}{index}{}%
  \glsnoundefn{indpart}{induced partition}{}%
  \glsnoundefn{relmsdpart}{msd relative to partition}{}%
  % Definition 2.12
  \label{defn:2.12}
  Let $G$ be a $3$-partite graph on $X \cup Y \cup Z$, and suppose that $G(X,
  Y)$, $G(X, Z)$, $G(Y, Z)$ are partitioned into $\bigcup_i G_i(X, Y)$,
  $\bigcup_j G_j(X, Z)$, $\bigcup_k G_k(Y, Z)$, respectively.
  For each $(x, y, z) \in \triangles(G)$, let its \emph{index} be the triple
  $(i, j, k)$ such that $xy \in G_i(X, Y)$, $xz \in G_j(X, Z)$, $yz \in G_k(Y,
  Z)$.

  The \emph{induced partition of $\triangles(G)$} is the partition of triples in
  $\triangles(G)$ according to their index.

  [A typical cell of this partition looks like $\triangles(G_i(X, Y) \cup G_j(X,
  Z) \cup G_k(Y, Z))$.]

  If $f : X \times Y \times Z \to [-1, 1]$, then the \emph{msd of $f$ relative
  to the partitions $\bigcup_i G_i(X, Y)$, $\bigcup_j G_j(X, Z)$, $\bigcup_k
  G_k(Y, Z)$} is defined to be the msd of $f$ relative to the induced
  partition.
\end{fcdefn}