Filling times for linear flow on the torus with truncated Diophantine conditions: a brief review and new proof

The paper’s Theorem 1 states that for δ ∈ (0, 1/2) and N* = (1 + n^2 n!)/δ, any direction α ∈ D^1_n(τ, γ, N*) produces a flow that fills T^n to within δ by time T < C(n, τ)/(γ δ^τ) with C(n, τ) = (1 + n^2 n!)^{τ+1}. The statement and its proof are given clearly and succinctly, using geometry-of-numbers (successive minima) to build a Z-basis {ω_j} with quantitative bounds and then choosing T accordingly; the steps and constants are explicit and consistent (see the statement and proof around Theorem 1 and its supporting proposition) , with notation and Diophantine sets defined earlier . The model’s (candidate) solution, while aiming for the same bound via a Fourier–positivity method, contains critical flaws: (1) it sets the trigonometric bandwidth Q = N* and then asserts the Fourier tail for ||k|| > N* vanishes; however, the construction enforces an ℓ∞ cutoff (||k||∞ ≤ Q), not an ℓ2 cutoff, so there are many modes with ||k|| > N* but ||k||∞ ≤ Q whose contributions do not vanish—this invalidates the truncation step needed to use only the Diophantine information up to N*; (2) it postulates one-dimensional nonnegative band-limited polynomials W_Q with simultaneously strong pointwise support, uniform ℓ^1 bounds on their Fourier coefficients, and weighted moment bounds scaling like Q^τ; these simultaneous requirements are not justified and are, in the stated form, implausible; and (3) it appeals to an unspecified unimodular change of variables with operator norm ≤ 1 + n^2 n! to reconcile Euclidean balls with slab intersections, without actually constructing or using it to fix the ℓ2/ℓ∞ mismatch. By contrast, the paper’s proof does not rely on these problematic steps and is internally coherent and complete. Therefore, the paper is correct and the model’s argument is not.