Recurrence rates for shifts of finite type

The paper’s Theorem 2.2 is clearly stated and proved with a careful multiscale construction that leverages the Gibbs property (quasi-Bernoulli), exponential correlation decay, and a nested-block scheme to control dependencies and guarantee a positive-measure set of sequences with the desired recurrence; exactness then upgrades this to full measure. See the statement of Theorem 2.2 and its context for Rψ in (2.5)–(2.6) and the assumptions (2.7)–(2.8) . The proof builds good-word sets G and the auxiliary sets B^{(ℓ)}_{i,δ}, C^{(ℓ)}_{i,δ}, proves a key measure estimate by exploiting exponential decay of correlations (Theorem 3.3) and a separated subfamily of return times (Lemma 5.2), and then iterates a nested construction D_k(i) to obtain a lower bound via an infinite product (Lemma 5.3), yielding μ([i]∩Sψ,(n_k)) ≥ c μ([i]) and hence μ(Sψ,(n_k))=1 and finally μ(Rψ)=1 . By contrast, the model’s solution relies on a purported “uniform finite-energy” bound and a first Borel–Cantelli argument. While its Step 1 lower bound μ([a]∩σ^{-p}[a]) ≳ μ([a])^2 is consistent with Gibbs/quasi-Bernoulli structure (2.2) , the crucial Step 3 claims a uniform conditional lower bound given the complement of earlier hits using only quasi-Bernoulli; this conditioning involves complements (non-cylinder events) and, without using correlation decay, is not justified. Most importantly, the model then asserts that summability in k follows merely because the exponent in its bound tends to infinity; (2.8) only guarantees divergence to +∞, not that the series of probabilities is summable, so the invoked first Borel–Cantelli step is invalid. The paper’s proof avoids this pitfall by a more delicate inductive construction that controls dependence across many windows using exponential mixing and a separated subfamily of return times, rather than requiring summable error terms .