On the number of flats spanned by a set of points in PG(d, q)

It is shown that for fixed 1 ≤ r ≤ s < d and > 0, if X ⊂ PG(d, q) contains (1+ )qs points, then the number of r-flats spanned by X is at least c( )q(r+1)(s+1−r), i.e. a positive fraction of the number of r-flats in PG(s + 1, q).