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

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