An exceptional set of uniformly spread Kakutani tilings of the line

TL;DR

This paper characterizes specific Kakutani tilings of the real line that are uniformly spread, revealing exactly five such cases linked to particular algebraic properties of the substitution parameter.

Contribution

It identifies the precise conditions under which Kakutani tilings are uniformly spread, connecting tiling properties with Pisot-Vijayaraghavan polynomials.

Findings

Exactly five values of alpha produce uniformly spread tilings.

Uniform spreadness is characterized by bounded displacement of a lattice.

The proof involves substitution tilings, Solomon's criterion, and algebraic classification.

Abstract

The {\alpha}-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - {\alpha}, for a fixed {\alpha} 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({\alpha}, 1 - {\alpha}) 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.