Reconstructing a Minimal Topological Dynamical System from a Set of Return Times

The paper’s Theorem 1.4 is proved by building the product orbit-closure Θ ⊂ X×K under S(x,k)=(Tx,k+α), defining H={h∈K:(x0,h)∈Θ}, showing H is a closed subgroup, then proving H⊂StabK(U′); since StabK(U′)={0}, H is trivial, so the X-projection is injective and yields the desired factor map φ with φ(Tx)=φ(x)+α and φ(x0)=0. This argument is complete and correct in the paper’s Section 2 (see the statement and set-up of Theorem 1.4 and its proof via Θ and H ⊂ StabK(U′) in the text) . By contrast, the candidate solution tries to force a unique K-fiber using an intersection of two syndetic sets; but two syndetic sets need not intersect (e.g., evens vs. odds), so the key step is invalid. The paper’s subgroup–stabilizer approach avoids this pitfall, while the model’s proof relies on a false combinatorial assumption, rendering it incorrect.