Grid method for divergence of averages

The paper proves strong sweeping out for products of rational powers with pairwise coprime denominators using a grid-method criterion (Proposition 3.8) plus a flow Conze principle, and then verifies the criterion via a multiplicative partition into b_i-free pieces and integer-power factors; its arguments are coherent and complete. The model follows the same high-level structure (grid method + Conze + Besicovitch) but proposes a different number-theoretic partition (an R-reduced, large-prime filtration). That partition appears plausible but is less explicitly justified in counting/density than the paper’s e_k S1 partition. Hence both establish the same theorem; the model’s proof is essentially the same framework with a different partition device.