Jop Bri\"{e}t \and Shravas Rao

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}.

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}.

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)

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

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

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

\[

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

\]

\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)

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

\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.

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