Incidence estimates for tubes

School of Mathematics Colloquium
Thursday, October 12, 2023 - 11:00am for 1 hour (actually 50 minutes)
Skiles 005
Hong Wang – NYU, Courant Insitute – hw3639@nyu.edu
Gong Chen

Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L,    I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?


The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least  (3s+1)/2 when s>0. 


We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture