This course is on arithmetic analogues in harmonic analysis with an outlook to pointwise ergodic theorems. I assume knowledge of the topics in my previous course from Spring 2019 discussing the basics of harmonic analysis.
Below is a tentative list of lectures.
- Stein's spherical maximal function and Jones's ergodic theorem
- Intro to the circle method: Waring's problem
- Bourgain's averages along the squares 1
- Bourgain's averages along the squares 2
- The discrete spherical maximal function of Magyar--Stein--Wainger
- Discrete analogues in harmonic analysis: Questions and future directions