Exponential Diophantine approximation and symbolic dynamics

The paper’s Theorem 2 explicitly states that for any monic, hyperbolic P(X) and any k ≥ 0, the set L^{(k)}(P) is closed in [0, 1/2], and the authors indicate it can be proved exactly as in their earlier work for Pisot numbers (see the statement and their proof pointer for Theorem 2). Their setup precisely defines L^{(k)}(P) via exponential–polynomial sequences and employs an ‘inverse formula’ intertwining to symbolic dynamics, which supports this conclusion (definitions and statement appear together with the inverse formula exposition). Hence, the paper’s claim aligns with its framework and prior methods, and I find no gap in the stated proof strategy. By contrast, the model’s compactness/torus argument contains a critical misstep in the final closedness step: it fixes a time t and derives a contradiction with limsup values across m, but a single fixed iterate does not control limsup as n→∞ for each m. This can be repaired by letting the witnessing times depend on m (choose t_m so that φ(T^{t_m}u) stays above the target and invoke continuity along the convergent u_m→u), but as written, the argument is invalid. Paper: Theorem 2 and its proof sketch are given; the inverse formula (mod Z realization) is developed to support such applications. Candidate: the flawed contradiction step uses a fixed t; the repair requires varying t_m. Citations: theorem and definitions (, ), the authors’ proof pointer “as in the Pisot case” (), and the inverse formula ().