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.