A Szemerédi Type Theorem for Sets of Positive Density in Approximate Lattices

The candidate solution proves the theorem by (i) a transverse correspondence principle for uniformly discrete sets, (ii) verifying a separation condition, (iii) a multiple-recurrence result for cross-sections along endomorphisms, and (iv) transferring back to upper Banach densities—exactly the route used in the paper to derive Theorem 1.5 from Theorems 3.1 and 1.9 (with Lemma 2.11) and the proof in Section 6. The core steps align with the paper’s Theorem 3.1 (Transverse Correspondence Principle) and Theorem 5.1/1.9 (multiple recurrence for cross-sections), culminating in Theorem 1.5’s combinatorial conclusion, with the same choice ∆ = Λ_q and the same inclusion Ξ_TPo − ∆ ⊂ Λ_{q+2} ensuring separation. Minor issues in the model solution are (a) a notational slip referring to the punctured hull of Λ instead of P_o, and (b) stating the correspondence lower bound for arbitrary finite T ⊂ G rather than T ⊂ ∆; these do not affect the argument where T = {α_k(λ)} ⊂ ∆ is all that is needed. Overall, both arguments match in structure and substance, and the paper’s steps are carefully justified in the cited results.