On strong forms of the Borel–Cantelli lemma and intermittent interval maps

The uploaded paper (Frolov, 2021) proves precisely the claim in question: for the intermittent map T_α on (0,1] with α∈(0,1), any sequence of intervals B_n contained in (d,1] with ∑ μ(B_n)=∞ satisfies the quantitative Borel–Cantelli (QBC) estimate S_n = E_n + o(E_n^{(1+α)/2}(log E_n)^{3/2}(log log E_n)^{(1+ε)/2}) μ-a.s., hence {B_n} is strong Borel–Cantelli (SBC) as a corollary. This is Theorem 3 and Corollary 2 of the paper, and its proof explicitly invokes a quantitative BC lemma (Theorem 1) plus a verification scheme (Theorem 2) and Gouëzel’s correlation bound for intervals in (1/2,1] (bound (1.3) there), exactly as summarized in the paper’s proof of Theorem 3 . By contrast, the candidate solution misstates key elements of Frolov’s framework. First, it incorrectly quotes the QBC lemma’s conclusion as S_n−E_n = o(g(E_n)(log E_n)^{3/2}(log log E_n)^{(1+ε)/2}); in the paper, Theorem 1 gives S_n−E_n = o(√(g(E_n)ψ(log E_n)(log E_n)^{3/2})) (with ψ chosen appropriately), and Corollary 1 is derived by selecting g(x)=Cx^{1+γ} and ψ(x)=(log x)^{1+ε} to obtain the log exponents stated in Theorem 3 . Second, the candidate asserts g(u) ≍ u^{(1+α)/2}; the paper’s verification (Theorem 2 with c(x) ≍ x^{1−1/α}) yields g(x)=x f^{-1}(x) ≍ x^{1+α}, whose square root gives the required E_n^{(1+α)/2} factor in the final error term . Third, the candidate’s pair-correlation control is misstated: Frolov uses Gouëzel’s two-term expansion in the form |μ(T^{-i}B_i ∩ T^{-j}B_j) − (1+c_{j−i})μ(B_i)μ(B_j)| ≤ C μ(B_j)/(j−i)^{1/α}, with c_{n}=O(n^{1−1/α}), leading to condition (6) with b_n≈n^{−1/α} (summable) and c_n≈n^{-(1/α−1)}; the candidate suppresses the μ(B_j) factor and replaces n^{−1/α} with n^{−(1/α−1)}, which would fail to be summable for α>1/2 and thus would not verify Theorem 2 in that range . Finally, the paper’s proof includes a propagation step (Lemma 1) and an induction over preimages to extend the result from (1/2,1] to any (d,1], which the candidate does not address . Therefore, while the candidate’s conclusion matches the paper’s main result, its chain of reasoning contains crucial inaccuracies and would not independently validate the theorem without correcting these points.