Validity of the key approximation lemma (Lemma 3.1) for all p > 1

Prove Lemma 3.1 for every p > 1; that is, show that for any ε > 0 there exist real trigonometric polynomials P and γ with integer spectrum such that: (i) the zeroth Fourier coefficient of P vanishes and the supremum norm of its Fourier coefficients is less than ε; (ii) γ is within ε of 1 in the A^p(T) norm; (iii) γ·P is within ε of 1 in the A^p(T) norm; and (iv) γ·S_l(P) is uniformly bounded in A^p(T) over all partial sums S_l.

Background

Lemma 3.1 is a central technical tool enabling the construction of Schauder frames of uniformly discrete translates in L^p(R) for the range p ≥ (1+√5)/2. The lemma provides approximants on the circle with precise Fourier-analytic control.

Establishing Lemma 3.1 for all p > 1 would remove the current restriction on p in Theorem 1.1 and yield the existence of such frames for the full range p > 1.