Statistics and Gap Distributions in Random Kakutani Partitions and Multiscale Substitution Tilings

The paper’s Theorem 2.3 states the exact asymptotic formulas for expected counts and volumes in the random substitution semi-flow F^Σ_t(S), with integrands built from C_Σ(x), D_Σ(x) and normalization v^T H_Σ 1, independent of the seed type. The paper proves this via weighted-walk asymptotics on the incommensurable graph G_Σ, a Tauberian argument (Theorem 3.1), and an identification of Q_Σ = 1 v^T / (v^T H_Σ 1) (Lemma 3.2) . The model’s solution derives the same formulas using a spine/Markov-renewal viewpoint: the child selection is volume-biased, increments are Y = log(1/vol child), the non-arithmetic (incommensurable) assumption ensures a stationary overshoot, and the limit kernel yields exactly the densities in the paper; integrating recovers the ‘in particular’ formulas using ∫ d_u = −u log u and ∫ c_u = 1 − u, matching H_Σ and S_Σ − V_Σ . Aside from a minor phrasing where the model informally says the last-edge rate is “proportional to vol(T)” (the rigorous limit weights are proportional to vol(T)·ℓ(T) via uniform overshoot within an increment), the two approaches are consistent. Hence both are correct, with different proofs.