Points on Lines in F

In this paper we study an incidence problem in finite geometry. Suppose we are given a set X which is the union of some number of lines in Fq . We choose a subset Y of X, such that for each line ` in X, at least half of its points are in Y . We show that |Y | is always at least some fraction of |X|. Using the polynomial method and degree reduction, it was previously known that such a statement holds for large and small |Y | (when |Y | ≥ q2 log q or |Y | ≤ q2/ log q). We close the gap by proving the statement for the remaining cases. We first note that such a statement holds for a set of points such that each point lies on at most two lines. We then show that there cannot be too many points with three or more lines lying on the zero set of a nonlinear polynomial, and use this to prove the statement in the remaining cases.