Infinite polynomial patterns in large subsets of the rational numbers

The paper proves the two target theorems for Q via a careful ergodic/topological argument, including new tools for Q-actions, and documents why analogous integer statements fail. The candidate solution relies on a “polynomial central sets theorem” to force the infinite pair-pattern inside a central translate, but key steps are unsubstantiated or false: (i) it incorrectly upgrades a piecewise syndetic cover to claim an entire translate A−t is central; (ii) it invokes an imprecise VIP/central-sets principle that does not justify realizing the all-pairs constraint {b_i, P(b_i)+b_j: i<j} while simultaneously ensuring b_i ∈ C; and (iii) if its method worked, it would also (incorrectly) yield the integer case, contradicting results surveyed and proved in the paper. The paper’s proofs are coherent and complete; the model’s proof is not.