SEMINAR: Counting Arithmetic Progressions in Subsets of Finite Fields and Rings

Guest:  Haydar Göral, İzmir Institute of Technology

Title: Counting Arithmetic Progressions in Subsets of Finite Fields and Rings (MATH)

Date/Time: 5 November 2025, 13:40

Location: FENS G035

Abstract: A fundamental result in combinatorics is Szemerédi's Theorem, which answers a famous conjecture of Erdős and Turán. It states that any subset of the integers with positive upper density must contain arbitrarily long arithmetic progressions. While this settles the question of existence, it naturally leads to a more quantitative one: How many such progressions are there? In this talk, we will explore this counting problem in the setting of finite algebraic structures. We will focus on finite fields and finite rings. Our main tool is to use character sums, including Gauss and Jacobi sums, and certain Weil estimates. We will present our work on counting k-term arithmetic progressions within the set of squares in a finite field. One of our key results is a precise asymptotic formula for this number, but with a significantly improved error term. We will also see how its optimal sharpness, connected to the Sato-Tate conjecture (a theorem now). Finally, time permitting, we will move from asymptotic counts to exact formulas. We will indicate how our methods allow us to determine the exact number of 3-term arithmetic progressions in certain subsets of some finite rings.

Bio: Haydar Göral obtained his BSc from İstanbul Bilgi University, his MSc from Koç University, and his PhD from Université Lyon 1. His research areas include number theory, model theory, combinatorics and theoretical computer science. He obtained BAGEP Award (2025)--Science Academy. Currently, he is working at İzmir Institute of Technology as an associate professor.