**Reading material**

__Book__: Additive Combinatorics by Tao and Vu

__Articles__:

- How a straightforward identity gives non-trivial bounds for many questions for large sets over finite fields. A point-line incidence identity in finite fields, and applications.
- How to do better than the completion method when bounding $$ \left| \sum_{a,b \in A} \chi(a-b)\right| $$ where $A$ is a relatively large subset of the prime field $\mathbb{F_p}$. On some double sums with multiplicative characters.
- The Balog-Szemerédi-Gowers theorem. New bounds in Balog-Szemerédi-Gowers theorem.

**Presentations**

- September 13: Ben, Kubra, Lori, and Zhaiming will present an infinite family of examples of subsets $A \subset \mathbb{F}_q$ of cardinality $q^{2/3}$ that satisfy $|(A-A)(A-A)| = (1+o(1)) \tfrac{q}{2}$.
- October 4: Alex, Matt, Noah, and Peter will present how to count solutions to $$ (a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8) \ , a_1, \dots, a_8 \in A \subset \mathbb{F}_q$$ using multiplicative characters and also how to use their estimate to obtain a lower bound on $|(A-A)(A-A)|$.