[proofplan]
The proof transfers the sparse function $f_N$, which is only bounded by the pseudorandom majorant $\nu_N$, to a dense bounded model $g_N: \mathbb Z/N\mathbb Z \to [0,1]$. The dense model has essentially the same average as $f_N$, so [Szemerédi's theorem](/theorems/4609) gives a positive lower bound for its $k$-term progression average. The counting part of the $k$-pseudorandom majorant hypothesis transfers that lower bound back from $g_N$ to $f_N$, producing the required constant and the stated $o(1)$ error.
[/proofplan]
[step:Introduce the finite groups and progression averages]
For each positive integer $N$, define the finite abelian group $G_N := \mathbb Z/N\mathbb Z$. For every function $h: G_N \to \mathbb R$, define its normalized average by
\begin{align*}
\mathbb E_{x \in G_N} h(x) := \frac{1}{|G_N|}\sum_{x \in G_N} h(x).
\end{align*}
For every function $h: G_N \to \mathbb R$, define its $k$-term arithmetic progression average $\Lambda_{k,N}(h)$ by
\begin{align*}
\Lambda_{k,N}(h) := \mathbb E_{x,r \in G_N} \prod_{j=0}^{k-1} h(x+jr)
= \frac{1}{|G_N|^2}\sum_{x \in G_N}\sum_{r \in G_N}\prod_{j=0}^{k-1} h(x+jr).
\end{align*}
The desired conclusion is therefore
\begin{align*}
\Lambda_{k,N}(f_N) \ge c(k,\delta)+o(1).
\end{align*}
[/step]
[step:Replace the sparse function by its dense bounded model]
By the definition of a [$k$-pseudorandom majorant](/page/Pseudorandom%20Majorant) in the theorem statement, applied to the sequence of functions $f_N: G_N \to [0,\infty)$ satisfying $0 \le f_N \le \nu_N$, there exist functions $g_N: G_N \to [0,1]$ such that
\begin{align*}
\mathbb E_{x \in G_N} g_N(x) = \mathbb E_{x \in G_N} f_N(x)+o(1)
\end{align*}
and
\begin{align*}
\Lambda_{k,N}(f_N)=\Lambda_{k,N}(g_N)+o(1).
\end{align*}
Since $\mathbb E_{x \in G_N} f_N(x) \ge \delta$ for all sufficiently large $N$, the first relation gives
\begin{align*}
\mathbb E_{x \in G_N} g_N(x) \ge \frac{\delta}{2}
\end{align*}
for all sufficiently large $N$.
[guided]
For each positive integer $N$, let $G_N := \mathbb Z/N\mathbb Z$. For every function $h: G_N \to \mathbb R$, define its $k$-term arithmetic progression average by
\begin{align*}
\Lambda_{k,N}(h) := \mathbb E_{x,r \in G_N} \prod_{j=0}^{k-1} h(x+jr).
\end{align*}
The function $f_N$ is not bounded by $1$; it is only bounded by $\nu_N$. The purpose of the pseudorandomness hypothesis is precisely to allow us to replace $f_N$ by a dense bounded model. Applying the [$k$-pseudorandom majorant](/page/Pseudorandom%20Majorant) hypothesis to the sequence $f_N: G_N \to [0,\infty)$ is legitimate because the theorem assumes $0 \le f_N \le \nu_N$. We obtain functions $g_N: G_N \to [0,1]$ satisfying
\begin{align*}
\mathbb E_{x \in G_N} g_N(x) = \mathbb E_{x \in G_N} f_N(x)+o(1)
\end{align*}
and
\begin{align*}
\Lambda_{k,N}(f_N)=\Lambda_{k,N}(g_N)+o(1).
\end{align*}
The first formula says that the dense model has asymptotically the same density as $f_N$. Since $\mathbb E_{x \in G_N} f_N(x) \ge \delta$ for all sufficiently large $N$, the error term $o(1)$ is eventually at least $-\delta/2$. Hence
\begin{align*}
\mathbb E_{x \in G_N} g_N(x) \ge \delta-\frac{\delta}{2}=\frac{\delta}{2}
\end{align*}
for all sufficiently large $N$. The second formula is the counting transfer statement: it says that $f_N$ and $g_N$ count $k$-term progressions with asymptotically the same normalized average.
[/guided]
[/step]
[step:Apply Szemerédi's theorem to the dense model]
By the [Varnavides quantitative form of Szemerédi's theorem on cyclic groups](/page/Szemeredi%27s%20Theorem), for the fixed integers $k \ge 1$ and density parameter $\delta/2>0$, there is a constant $c_0(k,\delta)>0$ such that every function $g: G_N \to [0,1]$ with
\begin{align*}
\mathbb E_{x \in G_N} g(x) \ge \frac{\delta}{2}
\end{align*}
satisfies
\begin{align*}
\Lambda_{k,N}(g) \ge c_0(k,\delta)+o(1),
\end{align*}
where the $o(1)$ term depends only on $k$ and $\delta$ as $N \to \infty$. Applying this to $g_N: G_N \to [0,1]$ gives
\begin{align*}
\Lambda_{k,N}(g_N) \ge c_0(k,\delta)+o(1).
\end{align*}
[guided]
The model $g_N$ is now in the dense setting: it is a function from $G_N$ to $[0,1]$, and its average is at least $\delta/2$ for all sufficiently large $N$. These are exactly the hypotheses needed for the [Varnavides quantitative form of Szemerédi's theorem on cyclic groups](/page/Szemeredi%27s%20Theorem) with density parameter $\delta/2$: the function is bounded between $0$ and $1$, the ambient group is the cyclic group $G_N$, and the density lower bound is uniform for all sufficiently large $N$. Therefore there exists a positive constant $c_0(k,\delta)>0$, depending only on $k$ and $\delta$, such that
\begin{align*}
\Lambda_{k,N}(g_N) \ge c_0(k,\delta)+o(1)
\end{align*}
as $N \to \infty$. The important point is that the constant does not depend on $N$ or on the particular function $g_N$; it is uniform over all dense bounded functions with density at least $\delta/2$.
[/guided]
[/step]
[step:Transfer the lower bound back to the original sparse function]
Combining the counting relation from the $k$-pseudorandom majorant hypothesis with the Szemerédi lower bound for $g_N$, we obtain
\begin{align*}
\Lambda_{k,N}(f_N)
&= \Lambda_{k,N}(g_N)+o(1) \\
&\ge c_0(k,\delta)+o(1).
\end{align*}
Define $c(k,\delta) := c_0(k,\delta)$. Since $c_0(k,\delta)>0$, this gives
\begin{align*}
\mathbb E_{x,r \in G_N} f_N(x)f_N(x+r)\cdots f_N(x+(k-1)r) \ge c(k,\delta)+o(1),
\end{align*}
which is the claimed estimate.
[guided]
The last step is to return from the dense model to the original sparse function. We already defined
\begin{align*}
\Lambda_{k,N}(h) := \mathbb E_{x,r \in G_N} \prod_{j=0}^{k-1} h(x+jr)
\end{align*}
for functions $h: G_N \to \mathbb R$. The [$k$-pseudorandom majorant](/page/Pseudorandom%20Majorant) hypothesis gave the counting-transfer relation
\begin{align*}
\Lambda_{k,N}(f_N)=\Lambda_{k,N}(g_N)+o(1),
\end{align*}
and the dense Varnavides-Szemerédi step gave
\begin{align*}
\Lambda_{k,N}(g_N) \ge c_0(k,\delta)+o(1),
\end{align*}
where $c_0(k,\delta)>0$ depends only on $k$ and $\delta$. Combining these two asymptotic estimates gives
\begin{align*}
\Lambda_{k,N}(f_N)
&= \Lambda_{k,N}(g_N)+o(1) \\
&\ge c_0(k,\delta)+o(1).
\end{align*}
Now define $c(k,\delta) := c_0(k,\delta)$. Since $c_0(k,\delta)>0$, the constant $c(k,\delta)$ is positive. Expanding the definition of $\Lambda_{k,N}(f_N)$ gives
\begin{align*}
\mathbb E_{x,r \in G_N} f_N(x)f_N(x+r)\cdots f_N(x+(k-1)r) \ge c(k,\delta)+o(1),
\end{align*}
which is exactly the lower bound asserted in the theorem statement.
[/guided]
[/step]
