Some Counterexamples to the Central Limit Theorem for Random Rotations

The paper proves two counterexamples to the √n-CLT for additive functionals of the ±α random rotation chain: (i) for a Liouville α there is an analytic observable with P(S_n/n^s>1)>1/24 infinitely often, and (ii) under infinitely many Diophantine approximants |α−p/q|≤c/q^γ (γ≥2), for any r<(γ−1)(1−s)−2 there is ϕ∈C^r with P(S_n/n^s>√2/32)>η infinitely often. These are Theorem 1 and Theorem 2 and their detailed inductive proofs (via special arcs G^η_q, rational approximations α_k, and carefully chosen time scales N_k) are internally consistent and complete in the manuscript , with the key steps laid out in Lemmas/estimates and the inductive constructions for both the Liouville and Diophantine cases (see the proof structure around (13)–(20) for Theorem 1 and (21)–(26) for Theorem 2) . By contrast, the model’s Phase-2 solution contains a fatal mismatch in Part (1): it chooses analytic Fourier weights a_j ≈ 2^{-j^2}e^{-σ m_j} and relies on the constraint n_j≲δ_j^{-2} with δ_j=|m_jα−ℤ| for Liouville α. For the explicit Liouville number α=∑10^{-j!}, one has δ_j≈m_j^{-j} (with m_j=10^{j!}), so even at the maximal allowed n_j, the main-mode contribution satisfies a_j n_j^{1−s}≲e^{-σ m_j} m_j^{2j(1−s)}→0, contradicting the model’s requirement (a_j/2)n_j^{1−s}≥2; moreover the claim “δ_j≤e^{-m_j^{3/2}}” is false for this α. Hence the model’s Part (1) construction cannot work as stated. The paper’s method avoids this obstruction by not coupling n_j to δ_j via a small-angle expansion; instead it uses near-rationality of α to keep orbits inside G-sets for N_k steps with positive probability, allowing N_k to be chosen arbitrarily large relative to the (exponentially small) Fourier weight and yielding the advertised bounds . Part (2) of the model roughly tracks the paper’s exponents, but lacks the careful subsequence/induction needed to balance earlier-mode tails, and omits the stage-wise constraints (25)–(26) that make the argument uniform; the paper’s version supplies those details . Proposition 1 and the Gordin–Lifšic framework for the positive CLT direction are also correctly stated and used in the paper to delineate the Diophantine regime .