Incidences between quadratic subspaces over finite fields

Abstract

Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is an odd prime power. A quadratic subspace $(W,Q)$ of $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ is called dot$_{k}$-subspace if $Q$ is isometrically isomorphic to $x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}$. In this paper, we obtain bounds for the number of incidences $I(\mathcal{K},\mathcal{H})$ between a collection $\mathcal{K}$ of dot$_{k}$-subspaces and a collection $\mathcal{H}$ of dot$_{h}$-subspaces when $h \geq 4k-4$, which is given by \[\left | I(\mathcal{K},\mathcal{H})-\frac{|\mathcal{K}||\mathcal{H}|}{q^{k(n-h)}}\right | \lesssim q^{\frac{k(2h-n-2k+4)+h(n-h-1)-2}{2}}\sqrt{|\mathcal{K}||\mathcal{H}|}. \] In particular, we improve the error term obtained by Phuong, Thang and Vinh (2019) for general collections of affine subspaces in the presence of our additional conditions.