Mean dimension theory in symbolic dynamics for finitely generated amenable groups

The paper proves, for subshifts over G=G1×G2 with deg(G2)=1 and the product metric d, that mdimM(X,G1,d)≤c1·htop(X,G), mdimH(X,{BS1(n)},d)≥c2·htop(X,G), and (when c1=c2=c) rdim(X,σ1,{BS1(n)},d,μ)=c·hμ(X,G). These appear as Theorem 2.1 and Theorem 2.2 in the manuscript and are established via a Bowen-ball/cylinder count for the upper bounds and a Lindenstrauss covering lemma plus a Fano-type mutual-information lemma for the lower bounds . The candidate solution presents essentially the same overall strategy: (i) a precise Bowen-ball/cylinder identity to control covering numbers; (ii) pattern-growth/entropy along Følner sets for the metric mean-dimension upper bound; and (iii) a rate–distortion coding argument with a Fano-type bound to get the matching lower bounds, then invokes known inequalities linking rdim±, mdimH, and mdimM. The only substantive difference is that the paper proves the mdimH lower bound directly via a covering lemma, while the candidate routes it through rdim− and the Lindenstrauss–Tsukamoto variational inequalities. Given the overlap in techniques and conclusions, both proofs are correct and substantially the same in spirit, with minor presentational differences.