On balance properties of hypercubic billiard words

The paper proves that for hypercubic billiard subshifts (dimension d+1, d≥2) the language is not balanced on factors, by: (i) translating factor-balancedness into bounded discrepancy and then into a continuous coboundary via Gottschalk–Hedlund (Proposition 24), (ii) showing H1_strong(Xθ; R) is not finitely generated when d≥2 (Theorem 25), and (iii) showing the asymptotically negligible subspace H1_an(Xθ; R) has finite dimension d (Theorem 27), hence some factor-counting class ψ_w falls outside H1_an, so w is not balanced; this is stated explicitly in the proof of Theorem 1 (Section 6.1) where they conclude: “there exists a finite word w such that the class of ψw does not belong to H1_an(Xθ,R) … By Corollary 24 we conclude that the word w is not balanced” . The candidate solution follows the same cohomological strategy: it uses Gottschalk–Hedlund to equate balancedness with a continuous coboundary for 1_[v]−μ([v]), invokes FHK02 for the non-finitely-generated H1_strong, and KS17 for the finite-dimensional asymptotically negligible subspace. The only extra step the candidate makes explicit—namely, that locally constant functions (hence cylinder indicators) generate the strong cohomology—is standard for subshifts and is implicitly used in the paper’s conclusion that some ψ_w-class must lie outside H1_an. Thus, both are correct and essentially the same proof.