  Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that
$A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at
least $\eta d$. This says that if we choose a random vertex $x\in A$ and a
random neighbour $y$ of $x$, then the probability that $y\in A$ is at least
$\eta$. The work of this paper was motivated by an attempt to obtain a
qualitative description of closed subsets of the Cayley graph $\Gamma$ whose
vertex set is $\mathbb F_2^{n_1}\otimes \dots \otimes \mathbb F_2^{n_d}$ with
two vertices joined by an edge if their difference is of the form $u_1\otimes
\cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a
description was obtained by Khot, Minzer and Safra, a breakthrough that
completed the proof of the 2-to-2 conjecture. In this paper, we formulate a
conjecture for higher dimensions, and prove it in an important special case.
Also, we identify a statement about $\eta$-closed sets in Cayley graphs on
arbitrary finite Abelian groups that implies the conjecture and can be
considered as a "highly asymmetric Balog-Szemer\'edi-Gowers theorem" when it
holds. We conclude the paper by showing that this statement is not true for an
arbitrary Cayley graph. It remains to decide whether the statement can be
proved for the Cayley graph $\Gamma$.
