Exploration on Incidence Geometry and Sum-Product Phenomena

Abstract

In additive combinatorics, Erd\"{o}s-Szemer\'{e}di Conjecture is an important conjecture. It can be applied to many fields, such as number theory, harmonic analysis, incidence geometry, and so on. Additionally, its statement is quite easy to understand, while it is still an open problem. In this dissertation, we investigate the Erd\"{o}s-Szemer\'{e}di Conjecture and its relationship with several well-known results in incidence geometry, such as the Szemer\'{e}di-Trotter Incidence Theorem. We first study these problems in the setting of real numbers and focus on the proofs by Elekes and Solymosi on sum-product estimates. After introducing these theorems, our main focus is the Erd\"{o}s-Szemer\'{e}di Conjecture in the setting of $\mathbb{F}_p$. We aim to adapt several ingenious techniques developed for real numbers to the case of finite fields. Finally, we obtain a result in estimating the number of bisectors over the ring $\mathbb{Z}/p^3\mathbb{Z}$ with $p$ a $4n+3$ prime.