Title: Discrete analogues of maximally modulated oscillatory integrals of Stein-Wainger type
Abstract: In 2001, Stein and Wainger introduced an interesting class of maximally modulated oscillatory integral operators related to Carleson's theorem. This talk is about discrete analogues for some of these operators. As is typical for a lot of discrete analogues, the corresponding discrete problem features a number of new and substantial difficulties arising from a curious fusion of number theory and analysis. In this talk we'll explore the theory for the one-dimensional discrete operator that was recently developed by Ben Krause (2018) in the special case of L^2 and present an application of recent variation-norm estimates for the continuous counterparts (Guo-Roos-Yung 2017) that allows us to obtain L^2 bounds for the discrete analogue in higher dimensions.