Incidences between Euclidean spaces over finite fields

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> {\mathbb{F}_{q}} be the finite field of order q , where q is an odd prime power. Then a k -dimensional quadratic subspace <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>W</m:mi> <m:mo>,</m:mo> <m:mi>Q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(W,Q)} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>dot</m:mi> <m:mi>𝐤</m:mi> </m:msub> </m:math> {\operatorname{dot}_{\mathbf{k}}} -subspace if Q is isometrically isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:math> {x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}} . In this paper, we obtain bounds for the number of incidences <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>I</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒦</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {I(\mathcal{K},\mathcal{H})} between a collection <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒦</m:mi> </m:math> {\mathcal{K}} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>dot</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> {\operatorname{dot}_{k}} -subspaces and a collection <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℋ</m:mi> </m:math> {\mathcal{H}} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>dot</m:mi> <m:mi>h</m:mi> </m:msub> </m:math> {\operatorname{dot}_{h}} -subspaces when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>h</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mn>4</m:mn> </m:mrow> </m:mrow> </m:math> {h\geq 4k-4} , which is given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo fence="true" maxsize="160%" minsize="160%">|</m:mo> <m:mrow> <m:mrow> <m:mi>I</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒦</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi mathvariant="script">𝒦</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> <m:msup> <m:mi>q</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mfrac> </m:mrow> <m:mo fence="true" maxsize="160%" minsize="160%">|</m:mo> </m:mrow> <m:mo>≲</m:mo> <m:mrow> <m:msup> <m:mi>q</m:mi> <m:mfrac> <m:mrow> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mn>4</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mi>h</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:msqrt> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi mathvariant="script">𝒦</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:msqrt> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> \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.