Infinite linear patterns in sets of positive density

The paper proves that for every A ⊆ ℕ of positive upper Banach density and for all weight vectors w1,…,wd ∈ {−r,…,r} with all nonzero prefix sums, there exist t and an infinite B with kB ⨹ w1B ⨹ ⋯ ⨹ wdB ⊆ A − t (Theorem 1.2), and gives a detailed ergodic-theoretic proof via a correspondence principle, pronilfactors, and a key positivity lemma (Lemma 3.2) culminating in an inductive construction of B . By contrast, the model’s solution hinges on the claim that a positive-density set admits a central translate, i.e., that some A − t lies in a minimal idempotent ultrafilter, and then invokes a Milliken–Taylor form of the Central Sets Theorem. This first step is incorrect: piecewise syndeticity (which follows from positive density) does not in general yield a central translate; if A − t were central for some t, then A would itself be central by translation-invariance of the notion, which is strictly stronger than piecewise syndetic. Hence the reduction to a central set C = A − t is unjustified. Even setting that aside, the model cites a version of the Central Sets Theorem allowing arbitrary signed compressed coefficient sequences without providing the necessary (nontrivial) justification, whereas the paper proves the uniformity over all admissible weights directly within its ergodic framework .