We discuss the necessary and sufficient condition for the boundedness of the

two-parameter exponential sum of e2πiP(t1,t2)t1t2.  By the application of the corresponding  discrete strong maximal average, we obtain the ergodic theorems regarding the rectangles whose side lengths have an arbitrary ratio. For the audience, I will illustrate the core idea of the circle method developed until now, as well as some history of the related problems. This is joint work with Hoyoung Song.