\end{lem}

Let $e: \mathbb{T} \rightarrow \mathbb{C}$ be the standard character $e(x) = e^{2\pi i x}$.  Then the inverse Gowers theorem states the following.

\begin{thm}[Inverse Gowers theorem~\cite{Tao:2012a}]

Let $\delta > 0$ and $s \geq 0$.  Then there exists an $\epsilon = \epsilon_{\delta, s}$ such that for every $\mathbb{F}_p^n$ and function $f : \mathbb{F}_p^n \rightarrow \mathbb{C}$ bounded in absolute value by $1$ such that $\|f\|_{U^{s+1}} \geq \delta$, there exists a non-classical polynomial $P: \mathbb{F}_p^n \rightarrow \mathbb{T}$ of degree $\leq s$  such that

\[

|\mathbb{E}_{x \in \mathbb{F}_p^n} f(x)e(-P(x))| \geq \epsilon.

\]

\end{thm}

Given the structure of non-classical polynomials that take on values in $\mathbb{T}$ described above, we have the following lemma.

\begin{lem}\label{lem:decomp}

Let $P: \mathbb{F}_p^n \rightarrow \mathbb{T}$ be a non-classical polynomial of degree $d < p$.  Then there exist non-classical polynomials $P_0, \ldots, P_{d}: \mathbb{F}_p^n \rightarrow \mathbb{T}$ of degree $d$ such that

\[

\sum_{j=0}^d P_j(x+jy) = P(y)

\]

for all vectors $x, y \in \mathbb{F}_p^n$.

\end{lem}

\snote{This is the lemma to change if we want to make our statement more general, I think}

\begin{proof}

Decompose $P$ into a sum of linear forms as in Lemma~\ref{lem:classtonon}.  For each $T_i$, we show that there exist $\alpha^i_0, \ldots, \alpha^i_{d}$ such that

\begin{equation}

\sum_{j=0}^d \alpha^i_j T_i(x+jy, \ldots, x+jy) = T_i(y, \ldots, y).\label{eq:need}

\end{equation}

We can then construct $P_j$ satisfying the lemma as follows.

\[

P_j(x) = \sum_{i=0}^d \alpha_j^i T_i(x).

\]

Because $T_i$ is an $i$-linear form, we can rewrite the left-hand side of~\eqref{eq:need} as

\[

\sum_{j=0}^d \sum_{s \in \{0, 1\}^i} \alpha^i_j T_i((1-s_1)x+js_1y, \ldots, (1-s_i)x+js_iy) = 

\sum_{j=0}^d \sum_{s \in \{0, 1\}^i} \alpha^i_j j^{|s|}T_i((1-s_1)x+s_1y, \ldots, (1-s_i)x+s_iy)

\]

where $|s|$ denotes the Hamming weight of $s$.  Then~\eqref{eq:need} holds if for all $0 \leq k < i$,

\[

\sum_{j=0}^d \alpha^i_j j^{k} = 0

\]

and

\[

\sum_{j=0}^d \alpha^i_j j^{i} = 1.

\]

This is a system of linear equations whose coefficients come from the first $i+1$ rows of a $(d+1) \times (d+1)$ Vandermonde matrix.  Because $d < p$, the determinant of such a matrix is non-zero and thus the matrix is invertible and $\alpha^i_j$ exist as desired.

\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:disctospec}]

 \begin{equation}

\end{equation}

  By the Lemma~\ref{lem:genvn},~\eqref{eq:unravel} is bounded above by $\|1_{H, K}\|_{U^t}$.  By the inverse Gowers theorem, there exists a polynomial $P$ of degree $t-1$ such that $\mathbb{E}_{h}[e(P(g)) 1_{H, K}] \geq \epsilon$ for some $\epsilon$.  By Lemma~\ref{lem:decomp}, there exist $P_0, \ldots, P_{t-1}$ such that \[e(P_0(x))e(P_1(x+y))\cdots e(P_{t-1}(x+(t-1)y)) = e(P(y)),\] and therefore \[\mathbb{E}_{u, g}[e(P_0(u))e(P_1(ug))\cdots e(P_{t-1}(ug^{t-1})) 1_{H, K}] = \mathbb{E}_{h}[e(P(g)) 1_{H, K}] \geq \epsilon,\]

and

\[

\]

\end{proof}

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