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\title{Blah Blah Blah}
\author{
       Jop Bri\"{e}t \and Shravas Rao
}
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\begin{document}

\title{Notes on hypergraph versions of Conlon--Zhao}
\maketitle

We attempt to prove that small discrepancy implies small spectral expansion for Cayley hypergraphs over certain types of abelian groups.  In the case of regular graphs, this is known to hold for Cayley graphs over any group by a result of Conlon and Zhao, but does not hold for general graphs~\cite{Conlon:2016}.

We start by giving definitions.  Because these are just my personal notes, I might reuse a lot of the definitions from my paper with Jop without stating them fully. In particular, let the definition of Cayley hypergraphs by the same, but let $\mathbf{g} = (0, g, g^2, \ldots, g^{t-1})$ and $q = \mathbf{1}$ be fixed for now.  In particular, $K = \cay^{(t)}(\Gamma, q, S)$ is a $t$-uniform Cayley hypergraph with edges induced by the set $S \subseteq \Gamma$, and we let $A_K$ be the corresponding adjacency tensor.  Given a Cayley hypergraph $K$, we say that a sub-hypergraph of $K$, $H = \cay^{(t)}(\Gamma, q, S')$ for some $S' \subseteq S$ has spectral expansion $\lambda$ with respect to $K$ if \[\lambda \geq \|A_H-A_K\|_{\ell_t, \ldots, \ell_t}.\]  We say that $H$ has discrepancy $\epsilon$ with respect to $K$ if for all $T_1, \ldots, T_t \subseteq \Gamma$, \[|A_H(1_{T_1}, \ldots, 1_{T_t})-A_K(1_{T_1}, \ldots, 1_{T_t})| \leq \epsilon |\Gamma|.\]  We can lower bound discrepancy in terms of the $\ell_{\infty}, \ldots, \ell_{\infty}$-norm of $A_H-A_K$ as stated in the following Lemma (the upper bound is trivial).  The proof follows almost exactly that of the same fact for matrices in Lemma 2.1 in~\cite{Conlon:2016}.

\begin{lem}\label{lem:disctoinfty}
If $A$ is an $n \times \cdots \times n$ $t$-linear form, there exist $T_1, \ldots, T_t \subseteq [n]$ such that \[\|A\|_{\ell_{\infty}, \ldots, \ell_{\infty}} \leq \pi^t|A(1_{T_1}, \ldots, 1_{T_t})|\]
\end{lem}

\begin{proof}
Let $\mathbb{D}$ be the set of complex numbers with absolute value $1$.  Recall that for all $z \in \mathbb{C}$ and a uniformly random unit complex vector $w$ that
\[
z = \pi \mathbb{E}_{w \in \mathbb{D}} [1_R(z \overline{w}) |z| w],
\]
where $R = \{z \in \mathbb{C} : \mathcal{R}(z) \geq 0\}$ is the set of all numbers with non-negative real part, and $1_R$ is the corresponding indicator function.  For a vector $x \in \mathbb{C}^{n}$, let $x'$ be the random vector defined by $x'_i = 1_R(x_i \overline{w})|x_i|$ where $w$ is again a uniformly random unit complex vector $w$.  Then we have
\begin{align}
\|A\|_{\ell_{\infty}, \ldots, \ell_{\infty}} &= \sup_{x[1], \ldots, x[t] \in \mathbb{D}^n} \sum_{i_1, \ldots, i_t \in [n]^t} A_{i_1, \ldots, i_t} x[1]_{i_1}\ldots x[t]_{i_t} \nonumber \\
&= \sup_{x[1], \ldots, x[t] \in \mathbb{D}^n} \left| \pi^t \mathbb{E}_{w_1, \ldots, w_t \in \mathbb{D}} \sum_{i_1, \ldots, i_t \in [n]^t} A_{i_1, \ldots, i_t} x'[1]_{i_1}\ldots x'[t]_{i_t} w_1 \ldots w_t \right| \nonumber \\
&\leq \pi^t \mathbb{E}_{w_1, \ldots, w_t \in \mathbb{D}} \sup_{x[1], \ldots, x[t] \in \mathbb{D}^n} \left| \sum_{i_1, \ldots, i_t \in [n]^t} A_{i_1, \ldots, i_t} x'[1]_{i_1}\ldots x'[t]_{i_t} \right|. \label{eq:reduce}
\end{align}
By convexity,~\eqref{eq:reduce} is maximized when $x'[j] \in \{0, 1\}^n$ for all $j$, or in other words, is an indicator vector.  Letting $T_j = \{i : x'[j]_{i} = 1\}$ proves the lemma.
\end{proof}

By a straightforward generalization of the Expander Mixing Lemma, it follows that if a Cayley hypergraph has spectral expansion $\lambda$, then it also has discrepancy $\lambda$.  We attempt to prove the converse, stated below.

\begin{lem}\label{lem:disctospec}
Let $\Gamma = F_p^n$. For $t \leq p$ and every $\lambda$ there exists an $\epsilon$ such that if a Cayley $t$-uniform hypergraph $H = \cay^{(t)}(\Gamma, q, S')$ has discrepancy $\epsilon$ with respect to $K = \cay^{(t)}(\Gamma, q, S)$, then it also has spectral expansion $\lambda$ with respect to $K = \cay^{(t)}(\Gamma, q, S)$.
\end{lem}

To prove this we use the generalized von Neumann inequality and the inverse Gowers theorem, which we state below.  We first state the definition of the Gowers norm.

\begin{define}
Given a function $f: \Gamma \rightarrow \mathbb{C}$, its Gowers uniformity norm is defined by
\[
\|f\|_{U^d} = [\mathbb{E}_{h_1, \ldots, h_d, x \in G} \Delta_{h_1}\cdots\Delta_{h_d} f(x)]^{1/2^d}
\]
where $\Delta_{h}$ is the multiplicative derivative
\[
\Delta_h f(x) = f(x+h)\overline{f(x)}.
\]
\end{define}

The following lemma relates the Gowers norm to \snote{not sure how to describe this}.  We state the form given in~\cite{Bhatt:2016}

\begin{lem}[Generalized von Neumann inequality]\label{lem:genvn}
Let $(\mathcal L_1, \ldots, \mathcal L_{t+1})$ be a collection of affine-linear forms $\mathcal L_i: (\mathbb{F}_p^n)^m \rightarrow \mathbb{F}_p^n$ for some prime $p$ such that no form is a multiple of another.  Then for $f_1, \ldots, f_{t+1}: \mathbb{F}_p^n \rightarrow \mathbb{C}$,
\[
|\mathbb{E}_{z_1, \ldots, z_m \in \mathbb{F}_p^n} \prod_{i=1}^{t+1} f_i(\mathcal L_i(z_1, \ldots, z_m))| \leq \min_{1 \leq i \leq t+1} \|f_i\|_{U^t}.
\]
\end{lem}

For certain groups $\Gamma$, a lower bound on the Gowers norm of a function $f$ implies certain structural properties of the function.  To do this, we define the notion of a non-classical polynomial.

\begin{define}
A non-classical polynomial of degree $< d$ is a function $f: \mathbb{F}^n \rightarrow G$ where $G$ is an additive group, if for all $h_1, \ldots, h_{d+1}, x \in \mathbb{F}^n$, the following holds.
\[
D_{h_1}\cdots D_{h_{d+1}} f(x) = 0
\]
where $D_h f(x) = f(x+h)-f(x)$.
\end{define}

We denote by $\mathbb{T}$ the circle group $\mathbb{R}/\mathbb{Z}$.  
%For any $\mathbb{F}_p$ there exists a natural homomorphism to $\mathbb{T}$ given by $(j) = j/p$. 
The following lemma gives an equivalent formulation of non-classical polynomials when the additive group is $\mathbb{T}$ and $d$ is less than $p$.

\begin{lem}[Lemma 1.7 in~\cite{Tao:2012a}]\label{lem:classtonon}
If $d < p$, then the non-classical polynomials of degree $d$ of the form $P: \mathbb{F}_p^n \rightarrow \mathbb{T}$ coincide with the classical polynomials of degree $d$.  In particular, $P$ can be expressed as
\[
P(x_1, \ldots, x_n) = \sum_{i=0}^d T_i(\mathbf{x}, \ldots, \mathbf{x})
\]
where $T_i$ is an $i$-linear form.
\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}]
We prove the contrapositive.  Assume that $\cay^{(t)}(\Gamma, q, S')$ does not have spectral expansion $\lambda$, that is that there exist vectors $x_1, \ldots, x_t$ such that $\|x_i\|_{\ell_t} = 1$ for all $i$, but $(A_H-A_K)(x_1, \ldots, x_t) \geq \lambda$.  Let $1_{H, K} = \left(\frac{1}{|S'|}-\frac{1}{|S|}\right)(1_{S'}-1_{S})$.  Then
 \begin{equation}
(A_H-A_K)(x_1, \ldots, x_t) = \frac{1}{t!} \sum_{\sigma \in S_t} \mathbb{E}_{u, g}[x_1(ug^{\sigma(1)-1})x_2(ug^{\sigma(2)-1})\cdots x_t(ug^{\sigma(t)-1})1_{H, K}(g)]. \label{eq:unravel}
\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
\[
(A_H-A_K)(e(P_0), \ldots, e(P_{t-1})) \geq \frac{\epsilon}{t!}.
\]
Because the coordinates of $e(P_i)$ are bounded above in absolute value by $1$ for all $i$, by Lemma~\ref{lem:disctoinfty}, this implies that $H$ has discrepancy at least $\epsilon/(t!\pi^t)$ with respect to $K$.
\end{proof}


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