Let $f : \mathbb{F}_q^l \rightarrow \mathbb{F}_q$ be a given function. We say that $f$ is a moderate expander if there exists an $\epsilon > 0$ such that for $|A| \gtrsim q^{1-\epsilon}$ one has that $|f(A,\ldots,A)| \gtrsim q$. We show that $x_1x_2 + (x_3-x_4)^2$, $f(x_1)+g(x_2) + x_3x_4$, $f(x_1) + g(x_2) + (x_3 - x_4)^2$, $f(x_1)g(x_2) + x_3x_4$, and $f(x_1) g(x_2) + (x_3 - x_4)^2$ are moderate expanders with the threshold $\epsilon = 3/8$ for various polynomials $f, g \in \mathbb{F}_q[x]$.

For a prime power $q$, we study the distribution of permanents of $n \times n$ matrices over a finite field $\mathbb{F}_q$ of $q$ elements. We show that if $\mathcal{A}$ is a sufficiently large subset of $\mathbb{F}_q$, then the set of permanents of $n \times n$ matrices with entries in $\mathcal{A}$ covers all (or almost all) of $\mathbb{F}_q^{*}$. When $q=p$ is a prime, and $\mathcal{A}$ is a subinterval of $[0,p-1]$ of cardinality $|\mathcal{A}| \gg p^{1/2} \log p$, we show that the number of matrices with entries in $\mathcal{A}$ having permanent $t$ is asymptotically close to the expected value.

Hart, Iosevich, Koh, and Rudnev [Averages over Hyperplanes, Sum-Product Theory in Vector Spaces over Finite Fields and the Erdös–Falconer Distance Conjecture, arXiv:0707.3473v2, 2007] show, using Fourier analysis method, that the finite Erdös–Falconer distance conjecture holds for subsets of the unit sphere in [Formula: see text]. In this article, we give a graph theoretic proof of this result.