Pointwise ergodic theorems for arithmetic sets

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.

1. Stein's spherical maximal function and Jones's ergodic theorem
2. Intro to the circle method: Waring's problem
3. Bourgain's averages along the squares 1
4. Bourgain's averages along the squares 2
5. The discrete spherical maximal function of Magyar--Stein--Wainger
6. Discrete analogues in harmonic analysis: Questions and future directions