Simple proofs of discretised projection theorems

We give a simple, short and self-contained presentation of Bourgain’s discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous dynamics. Our main innovation is a short elementary argument that shows that a discretised subset of $\mathbb{R}$ satisfying a weak ``two-ends’’ spacing condition is expanded by a polynomial to a set of positive Lebesgue measure.