On Burgess bounds and superorthogonality
Abstract: The Burgess bound is a well-known upper bound for short multiplicative character sums, with a curious proof. It implies for example a subconvexity bound for Dirichlet L-functions. In this talk we will present two types of new work on Burgess bounds. First, we will describe new Burgess bounds in multi-dimensional settings. Second, we will present a new perspective on Burgess’s method of proof. Indeed, in order to try to improve a method, it makes sense to understand the bigger “proofscape” in which a method fits. The Burgess method hasn’t seemed to fit well into a bigger proofscape. We will show that it can be regarded as an application of “superorthogonality.” This perspective turns out to unify topics ranging across harmonic analysis and number theory. We will survey these connections, with a focus on the number-theoretic side.