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

two-parameter exponential sum of $$extract_itex$$e^{2\pi i P(t_1,t_2)}{t_1t_2}$$/extract_itex$$.  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.