Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$

Abstract

For $E \subset \mathbb F_q^d$, $d \ge 2$, where $\mathbb F_q$ is the finite field with $q$ elements, we consider the distance graph $\mathcal G^{\textrm{dist}}_t(E)$, $t\neq 0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $\|x-y\| \equiv (x_1-y_1)^2+\dots+(x_d-y_d)^2=t$. We prove that if $|E| \ge C_k q^{\frac{d+2}{2}}$, then $\mathcal G^{\textrm{dist}}_t(E)$ contains a statistically correct number of cycles of length $k$. We are also going to consider the dot-product graph $\mathcal G^{\textrm{prod}}_t(E)$, $t\neq 0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x\cdot y \equiv x_1y_1+\dots+x_dy_d=t$. We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function $x\cdot y$ is not translation invariant. The exponent $\frac{d+2}{2}$ is improved for sufficiently long cycles.