Incidences between planes over finite fields

TL;DR

This paper establishes bounds on the number of incidences between k-planes and h-planes in finite fields using spectral graph theory, generalizing previous results and providing explicit bounds based on the sizes of the sets involved.

Contribution

The paper extends incidence bounds between planes in finite fields to more general cases, employing spectral graph theory techniques to derive new quantitative estimates.

Findings

Derived explicit bounds on incidences between k-planes and h-planes in finite fields.

Generalized previous incidence results to broader configurations with h ≥ 2k+1.

Provided a mathematical framework for analyzing geometric incidences in finite field vector spaces.

Abstract

We use methods from spectral graph theory to obtain bounds on the number of incidences between $k$-planes and $h$-planes in $\mathbb{F}_q^d$ which generalize a recent result given by Bennett, Iosevich, and Pakianathan (2014). More precisely, we prove that the number of incidences between a set $\mathcal{P}$ of $k$-planes and a set $\mathcal{H}$ of $h$-planes with $h\ge 2k+1$, which is denoted by $I(\mathcal{P},\mathcal{H})$, satisfies \[\left\vert I(\mathcal{P},\mathcal{H})-\frac{|\mathcal{P}||\mathcal{H}|}{q^{(d-h)(k+1)}}\right\vert \lesssim q^{\frac{(d-h)h+k(2h-d-k+1)}{2}}\sqrt{|\mathcal{P}||\mathcal{H}|}. \]