Discrete analogues in harmonic analysis: Multi-parameter Radon averages

In this paper we study maximal and oscillation inequalities for multi-parameter discrete Radon averaging operators. We develop a robust variant of the multi-parameter circle method within the framework of Discrete Analogues in Harmonic Analysis. In particular, this gives quantitative estimates for these averages and their underlying Fourier multipliers which reveals an interesting major-arcs rigidity phenomenon. As a consequence, we completely resolve in the affirmative the multi-parameter Bellow–Furstenberg problem in pointwise ergodic theory.