Problem: Let $p$ be an odd prime number and consider the $n$-dimensional vector space over the field with $p$ elements. I want to prove that the number of $3$-term arithmetic progressions in a subset $A\subseteq\mathbb{F}_{p}^{n}$ is $cp^{2n}$ for every $n\geq N_{0}$, for sufficiently large $N_{0}$ and a constant $c>0$.
Fact 1: I shall use Meshulam's theorem which states that if a subset $B\subseteq\mathbb{F}_{p}^{n}$ has density bigger than $2/n$, i.e. $|B|> \frac{2p^{n}}{n}$, then $Β$ contains a $3$-term arithmetic progression.
Fact 2: I also may use the following fact: First observe that if $U$ is a subspace of $\mathbb{F}_{p}^{n}$, then there are $p^{n-k}$ distinct cosets of $U$. Now, let $A\subseteq\mathbb{F}_{p}^{n}$ be a subset of density $\epsilon>0$. Then, there are at least $\frac{\epsilon}{2}p^{n-k}$ cosets $V$ of $U$ such that $|A\cap V|\geq \frac{\epsilon}{2}p^{k}$.
As an affine subspace of $\mathbb{F}_{p}^{n}$ is just a coset of a subspace of $\mathbb{F}_{p}^{n}$, we have that the above implies that there exist at least $\frac{\epsilon}{2}f(k,n)$ $k$-dimensinal affine subspaces $V$ of $\mathbb{F}_{p}^{n}$ such that $|A\cap V|\geq \frac{\epsilon}{2}p^{k}$, where $f(k,n)$ is the total number of the $k$-dimensional affine subspaces of $\mathbb{F}_{p}^{n}$. This number equals to \begin{equation} f(k,n)=p^{n-k}\binom{n}{k}_{p} \end{equation} where $\binom{n}{k}_{p}$ is the Gaussian coefficient and id defined as \begin{equation} \binom{n}{k}_{p}=\frac{(p^{n}-1)(p^{n}-p)\cdots(p^{n}-p^{k-1})}{(p^{k}-1)(p^{k}-p)\cdots (p^{k}-p^{k-1})}. \end{equation}
My attempt goes as follows: Let $n$ be large enough so that there exists $k<n$ such that $\frac{\epsilon}{2}>\frac{2}{k}$. Then, we consider $p^{n-k}$ $k$-dimensional affine subspaces $V$ of $\mathbb{F}_{p}^{n}$ pairwise disjoint (we can take such by taking the $p^{n-k}$ of a $k$-dimensional subspace). We know that there $\frac{\epsilon}{2}p^{n-k}$ of them such that $|A\cap V|\geq \frac{\epsilon}{2}p^{k}>\frac{2p^{k}}{k}$. Then, considering each $A\cap V$ as a subspace of $V\cong\mathbb{F}_{p}^{n}$, Meshulam's theorem implies that each one of them contains a $3$-AP and since they are pairwise disjoint there are $\frac{\epsilon}{2}p^{n-k}$ $3$-AP's in $A$.
That is as far as I get. My idea is to use the more powerful statement of Fact 2, where by finding $cp^{2n}$ disjoint $k$-dimensional affine subspaces of $\mathbb{F}_{p}^{n}$ it implies the desired. But I have no clue how to find them. Any ideas?