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In her book Quasicrystals and geometry Majorie Senechal discusses the relationship between quasicrystals and PV number on pp. 126-128. She cites Pisot (1946) and Cassels (1965).

First she presents an alternative characterization of PV numbers:

Theorem 4.1 Let $\mu_1>1$ be a real algebraic integer. Then $\mu_1$ is a PV number if and only if there exist nonzero $q \in \mathbb{R}$ such that $$\lim_{n\rightarrow \lim_{m\rightarrow \infty} \mu_1^m q = 0 \mod \mathbb{Z}.$$

Thus it follows that

Theorem 4.2 The diffraction condition is satisfied for an $\mathcal{A}$ sequence if and only if the leading eigenvalue $\lambda_1$ of $\mathcal{A}$ is a PV number.

I note that she defined an $\mathcal{A}$ sequence to be "any sequence of points $\Lambda = \{x_n\}$ such that, for all $n,$ $x_n - x_{n-1} \in \{\alpha_1, \ldots, \alpha_n\}$ and $\Lambda$ has suitably defined predecessors of all orders with respect to the [linear map which may be represented by a primitive matrix] $\mathcal{A}.$"

It's a nice book, easy to read. Highly recommended.

Oh, the references are:

C. Pisot (1946), Repartition (mod 1) des puissances successives des nombres reels, Commentarii Mathematici Helvetici, Vol. 19, 153-60.

J.W.S. Cassells (1965), Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics No. 45, Combridge University Press.

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In her book Quasicrystals and geometry Majorie Senechal discusses the relationship between quasicrystals and PV number on pp. 126-128. She cites Pisot (1946) and Cassels (1965).

First she presents an alternative characterization of PV numbers:

Theorem 4.1 Let $\mu_1>1$ be a real algebraic integer. Then $\mu_1$ is a PV number if and only if there exist nonzero $q \in \mathbb{R}$ such that $$\lim_{n\rightarrow \infty} \mu_1^m q = 0 \mod \mathbb{Z}.$$

Thus it follows that

Theorem 4.2 The diffraction condition is satisfied for an $\mathcal{A}$ sequence if and only if the leading eigenvalue $\lambda_1$ of $\mathcal{A}$ is a PV number.

I note that she defined an $\mathcal{A}$ sequence to be "any sequence of points $\Lambda = \{x_n\}$ such that, for all $n,$ $x_n - x_{n-1} \in \{\alpha_1, \ldots, \alpha_n\}$ and $\Lambda$ has suitably defined predecessors of all orders with respect to the [linear map which may be represented by a primitive matrix] $\mathcal{A}.$"

It's a nice book, easy to read. Highly recommended.

Oh, the references are:

C. Pisot (1946), Repartition (mod 1) des puissances successives des nombres reels, Commentarii Mathematici Helvetici, Vol. 19, 153-60.

J.W.S. Cassells (1965), Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics No. 45, Combridge University Press.