TOPOLOGICAL MULTIPLE RECURRENCE OF WEAKLY MIXING MINIMAL SYSTEMS FOR GENERALIZED POLYNOMIALS

The paper’s Theorem 1.1 states exactly the target claim, including the moreover clause that N ∩ C(ε, g1, …, gt) is syndetic, and proves it via a dedicated PET-induction for generalized polynomials together with Nils Bohr0-set regularizations (Lemmas 2.12, 2.16, 2.17) and a degree-1 base case (Lemma 3.1; Theorems 3.2–3.3), culminating in Theorem 4.1 and the full result (Section 4) . In contrast, the model’s solution relies on two critical missteps: (i) an over-strong “regularization” that replaces integer-valued generalized polynomials by integral polynomials Qi,r on a thick window and residue class—this fails already for p(n)=⌈αn⌉ with irrational α; and (ii) importing measure-theoretic weak-mixing PET for integral polynomials from a presumed weakly mixing invariant measure, which is not supplied by topological weak mixing alone. The paper avoids both issues by a self-contained topological PET framework and Nils Bohr0-set machinery that guarantee the required syndeticity directly .