New discretised polynomial expander and incidence estimates

TL;DR

This paper introduces new discretised polynomial expanders and incidence estimates using recent incidence geometry developments, with applications to discretised Elekes-Rónyai problems and non-concentration incidence scenarios.

Contribution

It provides novel discretised polynomial expanders and incidence bounds, advancing the understanding of geometric incidences under discretisation and non-concentration conditions.

Findings

Discretised version of Elekes-Rónyai expander problem established.

New incidence estimates for tubes and squares with non-concentration assumptions.

Applications demonstrate improved bounds in geometric incidence problems.

Abstract

We present two applications of recent developments in incidence geometry. One is a $\delta$-discretised version of a particular `Elekes--R\'onyai' expander problem. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.