THE SPECTRUM OF THE EXPONENTS OF REPETITION

The uploaded paper (Sim, 2021) proves exactly the two targets audited here. First, for slopes with bounded partial quotients, Theorem 3.3 states min L(θ) = lim_{k→∞}[1; 1 + a_k, a_{k-1}, …, a_1], with a constructive locating-chain argument that attains the bound (, ). Second, for the Fibonacci slope φ = [0;1̄], the paper identifies µmax = 1+φ, µ2, µ3, µ4, and µmin = 1+φ^2 (see the abstract and Figure 1), and proves: (i) µmin ≤ rep(x) ≤ µmax with sharp equality characterizations via eventual a- or b-chains and arbitrarily long chains, respectively (Theorem 4.4) (, ); (ii) the intervals (µ2, µmax) and (µ3, µ2) are maximal gaps, with endpoint characterizations by locating chains uab and vb2a2 (Theorem 4.5) (, ); and (iii) (µ4, µ3) is a maximal gap and µ4 is a limit point, realized by chains u(b^2a^2)^{e_1}ba(b^2a^2)^{e_2}ba… with lim sup e_i = ∞ (Theorem 4.6) (). The candidate solution reproduces these statements and employs the same core machinery (standard words M_k, convergents q_k, locating chains, and return-word/scale analysis). It even states the same constants and characterizations, e.g., min L(θ) via the reversed continued fraction, µmin = 1+φ^2, µmax = 1+φ, and the gap structure with the endpoint chains, matching Sim’s results. One minor textual inconsistency in the paper shows “uba” for µ2 in Proposition 4.6.1 while Theorem 4.5 uses “uab” (, ); the candidate follows Theorem 4.5’s ordering. Overall, both the paper and the model’s solution are correct and use substantially the same proof strategy.