Incidence Estimates for Tubes in Complex Space

TL;DR

This paper establishes a complex analogue of a tube incidence estimate under specific spacing conditions and applies it to a discretized Falconer distance problem in complex two-dimensional space.

Contribution

It introduces a new complex incidence estimate for tubes and uses it to solve a discretized Falconer distance set problem in ^2.

Findings

Proved a complex incidence estimate for tubes with strong spacing conditions.

Resolved a discretized Falconer distance set problem in ^2.

Extended real incidence geometry results to complex space.

Abstract

In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.