Monochromatic Sums and Products with Additive or Multiplicative Shifts in Natural Numbers

The paper rigorously proves Theorems 1.6 and 1.7. For Theorem 1.6, it uses a dynamical construction with a minimal idempotent in (βN,+) to select λ, a finite A with k-AP and k-FS, and a central B so that λA, λB, λ(A+B), and AB all lie in a single color class; see the setup with σs on X, the construction of Ek and the key identity establishing that λa has the target color, followed by the choice B = EN ∩ C and the verification that λ(A+B) and AB share the same color . For Theorem 1.7, it leverages a multiplicative minimal idempotent whose members are simultaneously central for + and ·, picks a color class D ∈ p + p, then builds E ∈ p and finite A, B ⊆ N so that (ρ+A) ∪ (ρ+B) ∪ (ρ+AB) ∪ (A+B) ⊂ D, establishing monochromaticity ; this uses the simultaneous centrality result (stated as Theorem 1.4 in the paper) and the formal statements of Theorems 1.6–1.7 are clear in the introduction . By contrast, the model’s outline leaves essential gaps: in Part A it tries to force AB ⊆ C using only additive centrality, invoking a finite-intersection property for the multiplicative preimages {b^{-1}C} without justification; in Part B it asserts simultaneous shrinking inside p ∩ q while preserving both additive and multiplicative head–tail structures, again without proof. These steps are precisely where the paper’s proofs avoid pitfalls by using dynamical recurrence (Part A) and an ultrafilter p with the strong simultaneous-centrality property plus the p + p mechanism (Part B).