Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

💡 Research Summary

The paper investigates the geometric relationship between separated nets (Delone sets) that arise from substitution tilings and the integer lattice ℤⁿ. Two notions of equivalence for separated nets are considered: (i) the existence of a bi‑Lipschitz bijection, and (ii) the existence of a bounded displacement (a bijection Φ with sup‖Φ(y)−y‖<∞). While it is known from work of McMullen and Burago–Kleiner that not every separated net in ℝⁿ is bi‑Lipschitz to ℤⁿ, the author shows that nets generated by primitive substitution tilings enjoy much stronger properties.

Main Results

1. Theorem 1.2 (2‑dimensional case).
   For any primitive substitution tiling of the plane, the associated separated net Y is bi‑Lipschitz equivalent to ℤ². The proof combines a quantitative estimate on the number of tiles inside large regions with the criterion of Burago and Kleiner (2002): if there exists a density α>0 such that the ratio between α·area(B) and the number of points of Y in a square B converges uniformly over scales, then Y is bi‑Lipschitz to ℤ². Using the substitution matrix A_H, the Perron–Frobenius eigenvalue λ₁ (which equals the inflation factor ξ²) and the second eigenvalue λ₂, the author shows that for any ε with 0<ε<λ₁−|λ₂| the discrepancy between tile count and area decays like δ^m where δ=(|λ₂|+ε)/λ₁<1. This exponential decay yields the required uniform convergence, establishing the bi‑Lipschitz equivalence.

2. Theorem 1.4 (Pisot case in any dimension).
   If the substitution H is primitive and Pisot (i.e., the second eigenvalue λ₂ satisfies |λ₂|<1), then for every separated net Y obtained from a tiling in the tiling space X_H there exists a constant β>0 and a bijection Φ:Y→β·ℤⁿ that is a bounded displacement. The argument relies on Laczkovich’s theorem (1992) which states that a separated net whose point‑count versus volume discrepancy is uniformly bounded admits a bounded displacement to a scaled lattice. The Pisot condition guarantees that the discrepancy term δ^m decays exponentially, making it uniformly bounded for all sufficiently large patches. Consequently, the net Y can be matched to a scaled integer lattice with only a uniformly bounded shift of each point.

3. Corollary 1.5 (Penrose tilings).
   The Penrose tiling is a classic example of a primitive Pisot substitution in ℝ². Applying Theorem 1.4 yields that any separated net derived from a Penrose tiling is a bounded displacement of β·ℤ² for some β>0, and therefore also bi‑Lipschitz to ℤ². This answers a question posed by Burago and Kleiner.

Technical Framework

• Substitution Tilings. A substitution H replaces each prototile by a finite patch of scaled copies (scale factor ξ>1). The substitution matrix B_H counts how many tiles of each type appear in the inflated image of a prototile; after collapsing congruent tiles, one obtains the smaller matrix A_H. Primitivity of H means that some power of A_H has all positive entries.

• Spectral Properties. By the Perron–Frobenius theorem, A_H has a simple dominant eigenvalue λ₁>1 with a strictly positive eigenvector v₁. The second eigenvalue λ₂ (in absolute value) governs the rate of convergence of tile frequencies to the limiting frequencies given by v₁. In the Pisot case, |λ₂|<1, which yields exponential convergence.

• Counting Estimates. Proposition 3.1 provides precise bounds for the number of tiles of each type inside a large patch P of a level‑m tiling τ_m, expressed in terms of t₁(m) (the number of tiles of the first basic type) and constants a₁, a₂ derived from v₁ and tile areas. The error term involves δ^m with δ as above.

• From Counting to Geometry. By embedding a point in each tile (any choice works because only the equivalence class matters), the tile count translates directly into the number of net points inside a region. The area of the region is approximated by the sum of tile areas, leading to the discrepancy estimates required by the Burago–Kleiner and Laczkovich criteria.

• Bounded Displacement Construction. Once the discrepancy is uniformly bounded, Laczkovich’s method constructs an explicit bijection Φ by matching points in Y to lattice points in a greedy, locally finite manner, ensuring that each point moves only a bounded distance.

Implications and Outlook

The results identify a broad class of separated nets—those arising from primitive (especially Pisot) substitution tilings—that are geometrically close to the integer lattice, both in the strong bi‑Lipschitz sense and the weaker bounded displacement sense. This contrasts sharply with the existence of pathological separated nets that cannot be so related to any lattice. The techniques blend combinatorial tiling theory, spectral analysis of substitution matrices, and deep results from geometric measure theory. Future work may explore non‑primitive or non‑Pisot substitutions, higher‑dimensional analogues, and quantitative bounds on the bi‑Lipschitz constants or displacement radii.