The frequency problem of the three gap theorem

The paper proves that, for Lebesgue-a.e. α, the asymptotic frequency of indices N with exactly two gap lengths is 0, i.e., lim_{n→∞}(1/n)#{1≤N≤n: #Δ_N(α)=2}=0, stated as Theorem 1 and reiterated in the conclusion (see Theorem 1 and its restatements and conclusion passages ). The proof proceeds by bounding the number of two-gap N in each interval determined by convergent denominators via Ravenstein-type structural facts (Theorem 3 and the “Scenario 1–3” counting arguments ), then using Lévy’s theorem for the exponential growth of q_n and a Borel–Bernstein/Borel–Cantelli lemma to ensure subexponential growth of partial quotients, yielding Lemma 6: (a_1+…+a_{n-1})/q_n→0 ( ). The candidate model solution proves the same main claim but via a sharper characterization (Slater) of exactly when the two-gap case occurs (N+1 equals a semiconvergent denominator), giving exact counts T(q_K−1)=∑_{j=0}^{K−1} a_{j+1}, and then the same Lévy + Borel–Cantelli metric inputs. Thus, both are correct; the paper uses upper bounds via Ravenstein’s recursion (“cake model”), whereas the model uses Slater’s exact classification. The conclusions agree while the proof techniques differ in their first step of counting.