Properties of Orbits and Normal Numbers in the Binary Dynamical System

The candidate solution reproduces the paper’s Theorem 1 bounds c1 Φ(n) ≤ (1/n)∑_{k=1}^n 1/f^{k-1}(x)^p ≤ c2 Ψ(n) with the same block-decomposition method, the same per-block estimates as Lemma 1 (zero-blocks give geometric 2^{pℓ} contributions) and Lemma 2 (one-blocks give arithmetic m contributions), and the same two-case decomposition depending on whether n falls in a zero- or one-block. The Φ, Ψ definitions match the paper’s (16)–(17), and the proof structure parallels equations (19)–(28), only with slightly more explicit constants and a direct handling of initial finite cases in place of the paper’s Lemma 3 smoothing. Nothing essential is missing: the solution assumes x ∈ Σ (no infinite zero/one runs), uses the left-shift property, and handles K0, J1, K1 consistently with the paper. Therefore, both arguments are correct and essentially the same. See Theorem 1 (18) with Φ,Ψ in (16)–(17), and Lemmas 1–2 (12),(15) for the core estimates; the case-by-case assembly corresponds to (19)–(28) .