**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.
- Polya-Vinogradov and Burgess using Fourier analysis.
- 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.
- A plan to combine the above two papers to prove non-trivial growth for products of differences.

**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 $|(A-A)(A-A)|$.
- October 18: Kubra and Lori will detail what we can infer about additive characters of the set studied in (1).
- October 25 Kubra will present a proof of Polya-Vinogradov.
- November 1 Peter and Alex will present a different proof of Polya-Vinogradov.
- November 8 Peter and Alex will present a proof of Burgess.
- November 29 a group attempt to materialize the plan listed above.