Book: Additive Combinatorics by Tao and Vu

Articles:

Presentations

1. 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 $|(A-A)(A-A)| = (1+o(1)) \tfrac{q}{2}$.
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 $|(A-A)(A-A)|$.
3. October 18: Kubra and Lori will detail what we can infer about additive characters of the set studied in (1).
4. October 25 Kubra will present a proof of Polya-Vinogradov.
5. November 1 Peter and Alex will present a different proof of Polya-Vinogradov.
6. November 8 Peter and Alex will present a proof of Burgess.
7. November 29 a group attempt to materialize the plan listed above.