Orders of Oscillation Motivated by Sarnak’s Conjecture, Part II

The paper proves that any oscillating sequence of order m = d + k − 1 is linearly disjoint from simple polynomial skew products of degree k on the d-torus, with f in triangular form (8) and k ≥ 2. The argument reduces to characters e(k·x), shows along any orbit the phase is a real polynomial in n of degree ≤ d + k − 1, then uses trigonometric approximation (Stone–Weierstrass) and the L^λ moment bound to control the error, completing the proof of Theorem 1. These ingredients are all present in the text: the exact form of f and statement of Theorem 1, the oscillation definition with the L^λ bound, the approximation scheme via trigonometric polynomials and Hölder control, and the key degree bound for the iterates that yields degree ≤ d + k − 1 for the character phases. The candidate solution follows the same structure: it reduces to characters, proves the same degree bound (via a discrete antidifference lemma instead of explicit summation), applies oscillation for monomials, extends to trigonometric polynomials, and then approximates continuous functions; it also uses the same L^λ moment bound to bound the uniform approximation error. Apart from a harmless notational slip in the base case of the degree bound, the model’s proof is correct and essentially the same as the paper’s proof in substance, with Fejér-kernel-based approximation replacing the paper’s Stone–Weierstrass step. See the theorem statement and form (8) and its proof and degree calculations in the paper, where the degree of the nth iterate’s i-th coordinate is bounded by i + k − 1, yielding the overall m = d + k − 1 and the trigonometric approximation and Hölder estimates used to pass from characters to general continuous observables.