Approximate incidence geometry in the plane

These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of δ δ -incidences Iδ(P,L) := \(p, ) ∈ P × L : p ∈ [ ]δ\ I δ ( P , L ) : = { ( p , ℓ ) ∈ P × L : p ∈ [ ℓ ] δ } under various hypotheses on P ⊂ R² P ⊂ R 2 and L ⊂ A(2,1) L ⊂ A ( 2 , 1 ) . The main focus will be on hypotheses relevant for the Furstenberg set problem .