An exceptional set of uniformly spread Kakutani tilings of the line

TL;DR

The paper classifies when Delone sets ${\Lambda_\alpha}$ arising from ${\alpha}$-Kakutani tilings are uniformly spread (BD-equivalent to a lattice). It separates into incommensurable and commensurable cases: incommensurable tilings exhibit large discrepancies, hence are not uniformly spread by Laczkovich’s criterion, while commensurable tilings reduce to primitive substitution tilings analyzed via Solomon’s criterion. A key step is showing the commensurable case corresponds to a fixed-scale primitive substitution with a characteristic polynomial $f_\alpha(x)=x^n-x^{n-m}-1$ whose roots determine BD behavior; the Pisot–Vijayaraghavan (PV) classification of these polynomials yields exactly five admissible values of $r_\alpha\in\{1,\tfrac{3}{2},2,3,4\}$, including the trivial lattice case $r_\alpha=1$. The four nontrivial values connect to well-known PV numbers (golden, plastic, supergolden, and the second PV) and their minimal polynomials, revealing a deep link between tiling dynamics, spectral theory, and number theory. The work also outlines extensions to more general multiscale tilings and higher dimensions, where PV-polynomial structures continue to influence uniform spread possibilities.

Abstract

The α-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - α, for a fixed α in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their associated Delone sets. We show that there are precisely five distinct values of min(α, 1 - α) for which these sets are uniformly spread, meaning that they are a bounded displacement of a lattice. The proof of this surprising fact combines the construction and analysis of a related family of primitive substitution tilings, Solomon's criterion for uniform spreadness, and a classification of Pisot-Vijayaraghavan polynomials.

Figures (6)

• Figure 1: The $\alpha$-Kakutani substitution rule on $I$.
• Figure 2: A patch of a $\alpha$-Kakutani tiling of ${\mathbb{R}}$, with $\alpha=1/3$.
• Figure 3: The graph associated with the $\alpha$-Kakutani substitution rule.
• Figure 4: The graph $G'_\alpha$ with $r_\alpha =\frac{3}{2}$ has $4=1+(3-1)+(2-1)$ vertices and edges of equal length $g_\alpha=\frac{1}{3}\log\frac{1}{\alpha}$.
• Figure 5: A primitive substitution rule $\rho_\alpha$ for the case $r_\alpha=\frac{3}{2}$.

Theorems & Definitions (25)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• Definition 3.1
• Definition 3.2
• Example 3.3