Coincidence Rank Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A one-dimensional self-similar inflation tiling uses finitely many labeled interval types: expanding each interval by $\lambda$ and subdividing it according to a substitution reproduces the tiling. Its substitution (abelianisation) matrix $M$ counts tiles of each type in each substituted tile. The translation hull $\Omega$ is the closure of its translates in the local tiling topology, with the translation-invariant probability measure of the substitution system. Pure point dynamical spectrum means that $L^2(\Omega)$ is spanned by eigenfunctions of the translation action. A Pisot number is a real algebraic integer $\lambda>1$ whose other algebraic conjugates have modulus less than $1$. The maximal equicontinuous factor is the largest factor of the translation system with an equicontinuous action. The coincidence rank $c$ is the almost-everywhere multiplicity of the factor map $\Omega\to\Omega_{\rm eq}$. The algebraic norm $N(\lambda)$ is the product of all conjugates of $\lambda$, an integer; divisibility uses $|N(\lambda)|$.

Statement recorded in the supplied source.

Does the coincidence rank of every one-dimensional Pisot inflation tiling divide $|N(\lambda)|$? [S2]

Short English statement

Is the multiplicity of the maximal equicontinuous factor always a divisor of the inflation factor’s algebraic norm?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600007 (AMR-065-0007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Faustin Adiceam (compiler), Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280v2 (2016), Section 2.2, Conjecture 2.2.3 and following definition, p. 4. https://arxiv.org/pdf/1604.06280.