INCIDENCES BETWEEN PLANES OVER FINITE FIELDS

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 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 K of k-planes and a set H of h-planes with h >= 2k + 1, which is denoted by I(K, H), satisfies vertical bar I(K, H) - vertical bar K vertical bar vertical bar H vertical bar/q((d-h) (k-1))vertical bar less than or similar to q (d-h h+k(2h-d-k+1)/2 root vertical bar K vertical bar vertical bar H vertical bar. As an application of incidence bounds, we prove that almost all k-planes, 1 <= k <= d - 1, are spanned by a set of 3(q)(d-1) points in F-q(d). We also obtain results on the number of t-rich incident k-planes and h-planes in F-q(d), with t >= 2.