). This improves on the previous best‐known bound, due to Jones.
To obtain our bound, we first prove an optimal point‐line incidence bound on Cartesian products, using a reduction to a point‐plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned.
We give several applications, to sum‐product‐type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.