Discrepancy and Rectifiability of Almost Linearly Repetitive Delone Sets

The paper proves exactly the target bound: for an ε-linearly repetitive Delone set Λ (with ε < r_Λ) there exist μ, α, δ > 0 such that for every axis-aligned box B, |#(Λ ∩ B) − μ·vol(B)| ≤ α·vol(B)/ℓ(B)^δ (Theorem 1.2) . It obtains this via a general uniform-averaging result for ε-weight distributions (Theorem 2.2) and then specializes to the counting weight w(B)=#(Λ∩B), which is an ε-weight distribution for any 0<ε<r_Λ (Proof of Theorem 1.2) . The proof develops a scale-contraction recursion for squarish boxes and extends to general boxes by partitioning into boxes of width ℓ(B), yielding the same decay rate in ℓ(B) (end of Theorem 2.2 proof) . The candidate’s solution follows the Lagarias–Pleasants scheme too—defining a local/weight distribution, proving a contraction across scales, and then passing from cubes to boxes—but implements two distinct proof devices: (i) a Vitali-type selection of many disjoint ε-copies across scales, whereas the paper takes a single interior copy plus a partition and derives a clean contraction on µ⁺−µ⁻ (cf. the inequalities culminating in Δ(C₂t) ≤ (1−C₁−C̄₁)Δ(t)+const·t⁻¹) ; and (ii) a Whitney-type cube decomposition, while the paper uses a simpler partition into boxes of side ℓ(B) to control the boundary term and sum the cubic estimates . Both arguments are sound and deliver the same discrepancy conclusion, hence the verdict.