Distribution of Integers with Digit Restrictions via Markov Chains

The paper proves a uniform distribution theorem (Theorem 4.8) for g-additive, eventually periodic functions on multiplicatively invariant sets represented by mixing, sofic, k-regular subshifts, under a precise Markov condition (irreducible and aperiodic Mi for i in one period). It constructs a family of doubly-stochastic transition matrices Mi (Definition 3.7, Proposition 3.8) whose stationary law is uniform on S = Z_a × V, shows exponential convergence under the Markov condition (Definition 4.2 and Lemma 4.3), and carefully passes from the ℓ-restricted language to all words (Proposition 3.10, Proposition 3.12, Corollary 4.1, and the α(ℓ,N) estimate culminating in Theorem 4.8). The logic is internally coherent and the steps are documented in the text . By contrast, the model’s proof outlines a Fourier-analytic diagonalization of a skew-product chain and a spectral gap argument that could, in principle, recover the same conclusion. However, it incorrectly identifies the character-summed count S_n(1) = 1^T T_{n-1}^{(≠0)} ∏_{i=0}^{n-2} T_i 1 with U_n, the number of length-n words with last digit nonzero. That expression counts labeled paths in the cover (possibly multiple paths bearing the same label across different start states), not distinct words. The paper explicitly avoids this pitfall by working in the ℓ-restricted language and fixing starting follower sets, where path–label counting is exact and grows like k^{n−ℓ} (Proposition 3.10) and by controlling the complement (Proposition 4.7 leading to the α(ℓ,N) bound used in Theorem 4.8) . Because the model proof never resolves this overcounting issue, its main identity and the subsequent division by U_n are not justified. With these corrections (adopting the paper’s ℓ-restricted setup or replacing the initial/final vectors by the correct initial distributions µ_i), the spectral approach could be repaired, but as written it is not correct.