A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics

The paper proves that for A-low-lying closed geodesics C ordered by period length, the normalized winding Ψ(C)/√ℓp(C) converges in distribution to a centered Gaussian with variance σp^2=(A^2−1)/12 (Theorem 1.3). It identifies Ψ(C) as the alternating sum of the partial quotients in the minimal even CF period and reduces the problem, via counting primitive necklaces and a characteristic-function argument, to a classical CLT for alternating sums of i.i.d. uniform digits; non-primitive contributions are shown negligible, and πA(N) is asymptotically cA·A^N/N so the mixture over n≤N is dominated by n near N, yielding the claimed limit (see Theorem 1.3, definitions of ΠA(N), πA(N), and the alternating-sum formula, as well as the counting and characteristic-function steps). The candidate solution reaches the same conclusion and variance using a different route: coding by a finite-alphabet shift (bounded digits), a two-state parity extension, and a periodic-orbit CLT for SFTs. This is compatible with the paper’s definitions and asymptotics. Hence both are correct, but the approaches differ substantially. Key touchpoints in the paper include the statement of Theorem 1.3, the identification Ψ(C)=a1−a2+…−an, the A^N/N counting of primitive classes, and the characteristic-function derivation of the Gaussian limit.