Incidence bounds related to circular Furstenberg sets

TL;DR

This paper establishes bounds on incidences between points and circles in the plane, leading to new lower bounds on the dimension of circular Furstenberg sets for large parameters.

Contribution

It introduces new incidence bounds between points and circles and derives novel lower bounds for the dimension of circular Furstenberg sets in the plane.

Findings

Established bounds on point-circle incidences in the plane.

Derived a new lower bound for the dimension of circular Furstenberg sets.

Extended results to large parameter regimes for Furstenberg sets.

Abstract

We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular $(u,v)$-Furstenberg sets, which is new for large $u$ and $v$.