An Incidence Result for Well-Spaced Atoms in all Dimensions

TL;DR

This paper establishes an incidence bound for well-spaced atoms and delta-tubes in all dimensions, aligning with heuristic predictions from the Szemerédi–Trotter Theorem, extending incidence geometry results.

Contribution

It provides a new incidence bound for well-spaced atoms and delta-tubes in all dimensions, matching heuristic predictions and generalizing previous results.

Findings

Incidence bound matches heuristic predictions

Applicable in all dimensions $d \\geq 2$

Extends incidence geometry theory

Abstract

We prove an incidence result counting the $k$-rich $\delta$-tubes induced by a well-spaced set of $\delta$-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemer\'edi--Trotter Theorem and holds in all dimensions $d \geq 2$.