Spanned lines and Langer's inequality

Abstract

We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was recently used by Han to improve the constant in the weak Dirac conjecture. Here we observe that this inequality also leads to improved constants in Beck's theorem, which states that a finite point set in the real or complex plane has many points on a line or spans many lines. Most of the proofs that we use are not original, and the goal of this note is mainly to carefully record the quantitative results in one place. We also include some discussion of possible further improvements to these statements.