Incidence Problems for Slabs

Abstract

In this thesis, I prove incidence estimates for slabs which are formed by intersecting small neighborhoods of well-spaced hyperplanes in R superscript 𝑑 with the unit cube [0, 1] superscript 𝑑 . My work is an analogue of a theorem of Guth, Solomon, and Wang, who proved a version of the SzemerédiTrotter theorem for thin tubes that satisfy a certain strong spacing condition. My proof uses induction on scales and the high-low method of Vinh, along with new geometric insights.