  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.
