A Dynamical View of Tijdeman’s Solution of the Chairman Assignment Problem

The paper’s Theorem 1.1 asserts exactly what the candidate proves: for totally irrational α there exist Tijdeman parameters producing a sequence u with Δα(u) ≤ 1 − 1/(2d−2) that is a bounded natural coding of the toral translation Tα by a partition into d finite unions of convex polytopes; the resulting subshift is minimal, uniquely ergodic, has pure point spectrum, and factor complexity of order n^{d−1}. The paper constructs the piecewise translation T̂α,C,C′ on V=1⊥ with atoms Sα,C,C′,i, establishes boundedness of the orbit (Proposition 3.8), builds a polytopal fundamental domain and natural partition conjugate to Tα (Proposition 3.9), derives discrepancy bounds (Proposition 3.10), and proves the Θ(n^{d−1}) complexity (Section 4) . The candidate’s solution mirrors this blueprint: same piecewise map on V with increments α−ei, same passage to the quotient V/Λ identifying all increments with α, same natural coding by polytopal atoms, same discrete-spectrum argument via being a factor of a toral rotation, and the same complexity growth via arrangements of O(n) hyperplanes. Minor differences are present in the model’s presentation (e.g., a brief “general position” heuristic for the lower complexity bound, and an algorithmic note on on-line generation), but these do not conflict with the paper. Hence both are correct and substantially the same proof.