Incidences between Euclidean spaces over finite fields

Abstract

Abstract Let 𝔽q{\\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 (𝔽qn,x12+x22+⋯+xn2){(\\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\\cdots+x_{n}^{2})} is called dot𝐤{\\operatorname{dot}_{\\mathbf{k}}}-subspace if Q is isometrically isomorphic to x12+x22+⋯+xk2{x_{1}^{2}+x_{2}^{2}+\\cdots+x_{k}^{2}}. In this paper, we obtain bounds for the number of incidences I\u2062(𝒦,ℋ){I(\\mathcal{K},\\mathcal{H})} between a collection 𝒦{\\mathcal{K}} of dotk{\\operatorname{dot}_{k}}-subspaces and a collection ℋ{\\mathcal{H}} of doth{\\operatorname{dot}_{h}}-subspaces when h≥4\u2062k-4{h\\geq 4k-4}, which is given by |I\u2062(𝒦,ℋ)-|𝒦|\u2062|ℋ|qk\u2062(n-h)|≲qk\u2062(2\u2062h-n-2\u2062k+4)+h\u2062(n-h-1)-22\u2062|𝒦|\u2062|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.