Asymptotically full measure sets of almost-periodic solutions for the NLS equation

The paper proves almost-periodic, Gevrey-regular global solutions to i u_t - u_xx + V * u + ε|u|^4 u = 0 for a positive-measure set of convolution potentials V and, for each such V, a positive-measure set of Gevrey initial data W, with measures asymptotically full as ε → 0, and it further constructs invariant (often embedded) tori and a Cantor foliation in a positive-actions region, relying on a counterterm theorem, an explicit tree expansion, and a bi-Lipschitz implicit-function approach in the parameters (V,W) (Theorem 2.3; Sections 5–8) . By contrast, the model’s solution outlines an infinite-dimensional KAM/Nash–Moser scheme with first and second Melnikov conditions and a twist obtained via Birkhoff normal form, which the paper explicitly seeks to avoid due to lack of twist; the paper’s non-resonance is phrased in terms of weak Diophantine sets and a counterterm compatibility system, not Melnikov conditions or quasi-Toeplitz KAM reducibility (Introduction and Sections 1.4–3) . The model also overclaims analytic submanifold structure of all invariant tori, whereas the paper proves submanifold structure only in a specific positive-actions region and notes failure of submanifoldness elsewhere (Section 8) . Hence, while some end conclusions sound superficially similar (existence and measure), the model’s proof framework does not match the paper’s assumptions or techniques and invokes unproven twist/Melnikov hypotheses absent from the paper.