Incidences between planes over finite fields

Abstract:

In this note, 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 generalizes a recent result given by Bennett, Iosevich, and Pakianathan (2014). More precisely, we prove that the number of incidences between a set $\mathcal {K}$ of $k$-planes and a set $\mathcal {H}$ of $h$-planes with $h\ge 2k+1$, which is denoted by $I(\mathcal {K},\mathcal {H})$, satisfies \[ \left \vert I(\mathcal {K},\mathcal {H})-\frac {|\mathcal {K}||\mathcal {H}|}{q^{(d-h)(k+1)}}\right \vert \lesssim q^{\frac {(d-h)h+k(2h-d-k+1)}{2}}\sqrt {|\mathcal {K}||\mathcal {H}|}. \]

As an application of incidence bounds, we prove that almost all $k$-planes, $1\le k\le d-1$, are spanned by a set of $3q^{d-1}$ points in $\mathbb {F}_q^d$. We also obtain results on the number of $t$-rich incident $k$-planes and $h$-planes in $\mathbb {F}_q^d$, with $t\ge 2$.