α-Kakutani Substitution Rule

• α-Kakutani Substitution Rule is a process that recursively splits a unit interval into two parts of lengths α and 1−α, generating substitution tilings and associated Delone sets.
• The resulting Delone sets are BD–equivalent to lattices only under five exceptional α-values, linking the arithmetic of α with uniform spatial spread.
• The theory employs primitive substitution techniques and spectral analysis, particularly Solomon’s criterion, to establish rigorous conditions for uniformity.

The α-Kakutani substitution rule is a parametric inflation–substitution process that splits the unit interval into two subintervals of lengths α and 1−α for a fixed α∈(0,1). Through repeated application, this rule generates substitution tilings of the real line and defines an associated family of Delone sets. The main structural property of these sets—whether they are uniformly spread, hence BD–equivalent to a lattice—depends in a subtle and rigid manner on the arithmetic properties of α and the associated spectral data of substitution matrices. Recent classification has identified exactly five exceptional values of min(α,1−α) for which the resulting Delone set is uniformly spread, thereby providing a comprehensive answer to the uniformity problem in this family (Smilansky, 28 Dec 2025).

1. Background: Delone Sets, Uniform Spreadness, and Discrepancy

Let ℝᵈ denote d-dimensional Euclidean space equipped with the standard norm and Lebesgue measure. A point set Λ⊂ℝᵈ is a Delone set if there exist 0<r≤R<∞ such that every open ball of radius r contains at most one point of Λ, and every closed ball of radius R contains at least one point of Λ, thus guaranteeing both uniform discreteness and relative density.

Two Delone sets Λ,Γ⊂ℝᵈ are BD–equivalent (Bounded Displacement equivalent) if there exists a bijection φ:Λ→Γ with sup_{x∈Λ}‖x−φ(x)‖<∞. A Delone set is uniformly spread if it is BD–equivalent to a full-rank lattice L⊂ℝᵈ; equivalently, if it is BD–equivalent to c·ℤᵈ for some c>0.

Given a Delone set Λ⊂ℝᵈ of asymptotic density d_Λ, the discrepancy over a bounded measurable region U and target density β is disc(Λ;β;U) := |#(Λ∩U) − β|U||. In one-dimensional space, Laczkovich’s criterion asserts that Λ is uniformly spread if and only if disc(Λ;d_Λ;I) is uniformly bounded over all intervals I. This links uniform spreadness to bounded discrepancy, a crucial concept for substitution tilings (Smilansky, 28 Dec 2025).

2. Primitive Substitution Tilings and Solomon’s Criterion

A primitive substitution tiling arises from a substitution rule ρ acting on a finite set of prototiles {T₁,…,T_k}, each homeomorphic (indeed biLipschitz) to a ball, and an inflation factor ξ>1. The rule ρ(T_j) specifies a finite patch made of translates of scaled tiles ξ⁻¹·T_i. The substitution matrix M_ρ records the number of each prototile within images of the substitution. Primitivity means some power of M_ρ is strictly positive.

Every substitution tiling induces a Delone set by associating one point per tile in a fixed biLipschitz manner; variations in the marking are BD–equivalent.

Solomon’s criterion provides a spectral condition for the uniform spreadness of such Delone sets: Let λ₁ (=ξᵈ),… ,λk be the eigenvalues of Mρ. For tilings in ℝᵈ, let ℓ be the minimal index ≥2 such that the corresponding eigenspace is not orthogonal to the all-ones vector. The Delone set Λ is uniformly spread if and only if |λ_ℓ| < λ₁^{(d–1)/d}. In dimension one (d=1), this reduces to: Λ is uniformly spread if and only if all eigenvalues other than λ₁ (Perron root) satisfy |λ|<1 (Smilansky, 28 Dec 2025).

3. The α-Kakutani Substitution Rule

For α∈(0,1), the α-Kakutani substitution recursively splits an interval into two pieces of lengths α and 1−α, inflates, and tiles the real line. This iterative process produces tiles whose lengths belong to {α^k(1−α)^ℓ: k,ℓ≥0}. The set of left endpoints of such tilings yields the Delone set Λα. The critical parameter rα := (log α)/(log(1−α)) divides the theory:

• Incommensurable case (r_α ∉ ℚ): The substitution graph has incommensurable loop lengths; the discrepancy of Λα on long intervals grows at least as fast as |I|/log|I|, violating Laczkovich’s bounded discrepancy criterion. Thus, Λα is never uniformly spread in this regime.
• Commensurable case (r_α∈ℚ; r_α=n/m with coprime n>m≥1): The substitution can be recast as a fixed-scale primitive substitution, with inflation ξ = α^{–1/n} on k = n+m–1 prototiles. The characteristic polynomial of the substitution matrix M_α has the form det(xI − M_α) = x^{n+m–1} − x^{m–1} − x^{n–1} = x^{m–1}(xⁿ − x^{n–m} − 1). The nonzero eigenvalues are exactly the roots of f_α(x) := xⁿ − x^{n–m} − 1.

Solomon’s criterion in d=1 applies: Λ_α is uniformly spread if and only if every non-Perron eigenvalue λ₂ satisfies |λ₂|<1 (Smilansky, 28 Dec 2025).

4. Exceptional Values and Classification via Pisot–Vijayaraghavan Polynomials

A thorough spectral analysis and classification rooted in Dubickas–Jankauskas’ results on Pisot–Vijayaraghavan (PV) polynomials reveals that only particular commensurable cases lead to uniform spreadness:

• None of the eigenvalues of f_α(x) other than the Perron root lie on the unit circle.
• |λ₂|<1 iff f_α(x) is irreducible and the minimal polynomial of a Pisot–Vijayaraghavan number.
• The only PV–polynomials of the special type xⁿ–x^{n–m}–1 are:

1. x²–x–1 (golden polynomial)
2. x³–x–1 (plastic ratio polynomial)
3. x³–x²–1 (supergolden)
4. x⁴–x³–1

Correspondingly, the only ratios r_α := n/m arise as 1, 3/2, 2, 3, 4. Including the trivial case r_α=1 (α=½), there are precisely five α-values (up to min{α, 1−α}) yielding BD–equivalence to a lattice:

r_α	Polynomial	α value	Interpretation
1	–	½	Periodic tiling
3/2	x³–x–1	≈0.430159…	Plastic ratio
2	x²–x–1	1/φ² ≈ 0.381966…	Golden ratio
3	x³–x²–1	≈0.317672…	Supergolden
4	x⁴–x³–1	≈0.275509…	Second smallest PV

Uniformly spreadness occurs if and only if r_α ∈ {1, 3/2, 2, 3, 4} (Smilansky, 28 Dec 2025).

5. Concrete Examples and Their Lattice Structure

A summary of the five exceptional cases:

1. r_α=1 (α=½): The interval splits equally; the resulting set is the integer lattice up to translation.
2. r_α=2 (α=1/φ²≈0.381966…): Associated to the golden ratio φ=(1+√5)/2; characteristic polynomial x²–x–1.
3. r_α=3/2 (α≈0.430159…): The plastic ratio case; f_α(x)=x³–x–1.
4. r_α=3 (α≈0.317672…): The “supergolden” ratio; f_α(x)=x³–x²–1.
5. r_α=4 (α≈0.275509…): Minimal polynomial x⁴–x³–1 for the second smallest PV number.

In all five, the substitutions yield Delone sets BD–equivalent to lattices; the second-largest eigenvalue of M_α lies strictly within the unit circle. For any other α, either the incommensurability or a non-PV polynomial for f_α preclude bounded discrepancy, leading to failure of lattice BD–equivalence (Smilansky, 28 Dec 2025).

6. Broader Context and Related Results

The α-Kakutani substitution exemplifies the interplay between symbolic dynamics, tiling theory, and Diophantine analysis. The only cases giving rise to uniformly spread Delone sets correspond to fundamental algebraic numbers, all connected to Pisot–Vijayaraghavan numbers, reflecting deep rigidity not present in generic substitution systems. Laczkovich’s and Solomon’s criteria underpinning these results demonstrate sharp connections among tiling spectrum, eigenvalue structure, and spatial uniformity.

These advances clarify longstanding questions about when substitution tilings can model regular discrete sets (lattices) up to bounded displacement and provide reference constructions for further investigations into substitution systems, quasi-crystals, and model sets (Smilansky, 28 Dec 2025).