Application of high-low method to distance problems

Abstract

In this thesis, I used the high-low method to study incidence problems arising form variants of the distance set conjectures.

In the complex space, my collaborator Sarah Tammen and I proved analogues of the incidence estimates of Guth, Solomon and Wang \cite{WellSpacedTubes} for tubes obeying strong spacing conditions, and we used one of our new estimates to resolve a discretized variant of Falconer's distance set problem in C².

In the real space, I proved an incidence estimate for special boxes in R⁶, and used the estimate to derive a bound for a discretized variant of Falconer's problem in R³.