Homological Pisot 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 $1$. Unimodular means $\det M=\pm1$. The rank of first rational Čech cohomology is $\dim_{\mathbb Q}\check H^1(\Omega;\mathbb Q)$.

Statement recorded in the supplied source.

Must a one-dimensional unimodular Pisot inflation tiling have pure point dynamical spectrum if $\dim_{\mathbb Q}\check H^1(\Omega;\mathbb Q)$ equals the algebraic degree of $\lambda$? [S2]

Short English statement

Does the stated minimal rational cohomology rank force a unimodular Pisot inflation system to have pure point spectrum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600006 (AMR-065-0006). 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 2.2.2, p. 4. https://arxiv.org/pdf/1604.06280.