Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

The paper proves Theorem A—dense simultaneous approximation for all points in minimal nilsystems and an almost-everywhere uniform version—via a structured argument using rational points and periodic polynomial orbits on product nilmanifolds, together with continuity properties of prolongations (Theorem 4.7 implies Theorem A) . Crucially, the a.e. result is obtained on a residual set Ω, not for all points, and whether the modulus N can be chosen uniformly for every x is explicitly posed as an open problem (Question 6.5) . The candidate solution asserts a stronger statement (uniform N for all x, i.e., Ω = X) and bases its induction on controlling only the Kronecker factor and a “constant” central fibre cocycle. The paper warns this approach is inadequate: the Kronecker factor is not characteristic for simultaneous approximation (Remark 4.4) , and the torus-style “near-identity after NL” tactic fails for general nilrotations—hence the need for periodic polynomial orbits and rationality machinery (Lemma 4.3 and the subsequent generalization; Theorem 3.14) . Therefore the model’s proof contains critical gaps and overclaims beyond what is known, while the paper’s argument is coherent and aligns with the state of the art.