Georgia Institute of Technology College of Sciences

Characters are a central tool for understanding arithmetic. For example, the most familiar character is the Legendre symbol, which detects the quadratic residues. In this talk I will present a few ideas as to how character sums may be useful in arithmetic combinatorics and vice versa. Traditionally, estimates for character sums have been used to count arithmetic configurations of interest to the combinatorialist. More recently, arithmetic combinatorics has proved useful in the estimation of certain character sums. Many character sums are easy to estimate provided they have enough summands - this is sometimes called the square-root barrier and is a natural obstruction. I will show how the sum-product phenomenon can be leveraged to push past this barrier.