# Relationship between quasicrystals and PV numbers

Freeman Dyson writes in his article "Birds and Frogs", that a unique quasicrystal exists corresponding to every Pisot-Vijayaraghavan number. Is this statement a theorem?

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+1 for relationsheep, birds, and frogs. –  Daniel Litt Jan 7 '11 at 19:09
This seems too strong a claim. In case they are useful, the following two links should give you some pointers on what is known: liafa.jussieu.fr/~cf/publications/tcs.ps (A paper: "Additive and multiplicative properties of point sets based on beta-integers" by Christiane Frougny, Jean-Pierre Gazeau & Rudolf Krejcar) and www.math.uni-bielefeld.de/~richard/ws08files/Gazeau.pdf (Slides: "Beta-lattice multiresolution of quasicrystalline Bragg peaks", by A. Elkharrat, J.P. Gazeau, and F. Denoyer.) –  Andres Caicedo Jan 7 '11 at 20:09
@Daniel, sorry, I edited the sheep out. –  Gerry Myerson Jan 8 '11 at 0:04
The first link above seems to have changed, and should be updated to liafa.univ-paris-diderot.fr/~cf/publications/tcs.ps –  Andres Caicedo Jun 20 '13 at 20:05

Do you have a particular definition in mind for quasicrystal? To my understanding, there are several related concepts that give different answers. If you want an expansion rule, such patterns exist in 2 dimensions for every algebraic integer that is larger than its Galois conjugates (and conversely, depending on requirements of the expansion rule). These are most easily constructed for PV numbers (a theorem I proved, cf Lecture notes by Thurston on tiling, or Richard Kenyon, http://www.math.brown.edu/~rkenyon/papers/sst.ps.Z).

One definition seems to be a low-dimensional slice of a higher-dimensional periodic arrangement, e.g. take an $n$-plane in $m$-space, look at all lattice points in a nbhd of radius r, and project. There are continuous families of these, and they are not associated with algebraic integers. These are said to have "long range translational order".

Either of these two constructions, and others as well, give ordered patterns in which matter in principle would "crystallize", if the energy of interactions of molecules could be carefully controlled.

The intersection of examples with expansion rules and long range translational order are the examples corresponding to PV numbers. I believe the locations of the sharp peaks in the X-ray diffraction diagram only depend on the PV number, and that's probably what Freeman Dyson had in mind --- I know he was very interested in X-ray diffraction.

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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 $$\lim_{m\rightarrow \infty} \mu_1^m q = 0 \mod \mathbb{Z}.$$

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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