ON AN ANALOGUE OF THE HUREWICZ THEOREM FOR MEAN DIMENSION

The paper proves that if the base has zero mean dimension then mdim(X,T) = mdim(π,T) (Theorem 1.5/4.9), by reducing to the small boundary property via a zero-dimensional free minimal extension and applying a relative mean-dimension comparison; see the statement and proof of Theorem 1.5/4.9 and Corollary 4.2 in the paper . The candidate’s solution establishes the obvious inequality mdim(π,T) ≤ mdim(X,T) and then simply invokes the paper’s Theorem 1.5 to get the reverse inequality, hence equality; this is correct, though the citation “this is precisely Proposition 1.1” for the easy inequality is slightly off since Proposition 1.1 identifies mdim(π,T) with the supremum of fiber mean dimensions rather than explicitly stating the global monotonicity, which the paper notes is obvious from the definition (see the start of the proof of Proposition 4.6) . The candidate also briefly conflates “zero-dimensional base” with “base of zero mean dimension,” but their Step 2 correctly uses the mdim(Y,S)=0 hypothesis as in the paper.