Incidences between Euclidean spaces over finite fields

Abstract

Abstract Let 𝔽 q {\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power. Then a k-dimensional quadratic subspace ( W , Q ) {(W,Q)} of ( 𝔽 q n , x 1 2 + x 2 2 + ⋯ + x n 2 ) {(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called dot 𝐤 {\operatorname{dot}_{\mathbf{k}}} -subspace if Q is isometrically isomorphic to x 1 2 + x 2 2 + ⋯ + x k 2 {x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}} . In this paper, we obtain bounds for the number of incidences I ⁢ ( 𝒦 , ℋ ) {I(\mathcal{K},\mathcal{H})} between a collection 𝒦 {\mathcal{K}} of dot k {\operatorname{dot}_{k}} -subspaces and a collection ℋ {\mathcal{H}} of dot h {\operatorname{dot}_{h}} -subspaces when h ≥ 4 ⁢ k - 4 {h\geq 4k-4} , which is given by | I ⁢ ( 𝒦 , ℋ ) - | 𝒦 | ⁢ | ℋ | q k ⁢ ( n - h ) | ≲ q k ⁢ ( 2 ⁢ h - n - 2 ⁢ k + 4 ) + h ⁢ ( n - h - 1 ) - 2 2 ⁢ | 𝒦 | ⁢ | ℋ | . \Bigl{\lvert}I(\mathcal{K},\mathcal{H})-\frac{\lvert\mathcal{K}\rvert\lvert% \mathcal{H}\rvert}{q^{k(n-h)}}\Bigr{\rvert}\lesssim q^{\frac{k(2h-n-2k+4)+h(n-% h-1)-2}{2}}\sqrt{\lvert\mathcal{K}\rvert\lvert\mathcal{H}\rvert}. In particular, we improve the error term in [N. D. Phuong, P. V. Thang and L. A. Vinh, Incidences between planes over finite fields, Proc. Amer. Math. Soc. 147 2019, 5, 2185–2196] obtained by Phuong, Thang and Vinh for general collections of affine subspaces in the presence of our additional conditions.