[Math] Mathematics of quasicrystals

Question

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.

Answer 1

The trouble with the quasicrystals is that the literature in this area is dominated by non-mathematical or pseudo-mathematical papers and books. In particular, just extracting a mathematical definition of a quasicrystal from this literature is not so easy. This situation is well-illustrated by the wikipedia article on quasicrystals and the MO discussion of this topic at What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?

The two papers that I found most enlightening and mathematical, address this problem head-on (more on this below):

[1]. A. Hof, "On diffraction by aperiodic structures", Commun. Math. Phys., 169 (1995), p. 25-43.

[2]. J-B. Gouere, "Quasicrystals and almost periodicity", Commun. Math. Phys., 255 (2005), p. 655-681.

Both papers prove some nontrivial mathematical theorems, on the basis of these theorems one can then form two, somewhat different, mathematical definitions of a (quasi)crystal. Senchal's 2-page long survey paper (see Mahmud's answer) or, better, her book (see Joseph's answer) is a good introduction, the trouble is that she does not prove anything in the book and that she could be sloppy with her definitions, for instance, she conflates functions, measures and distributions, which are needed for defining crystals.

If you look in Senchal's book, you first get the following physical definitions of crystals: "A crystal is any solid with essentially discrete diffraction diagram." (This includes both traditional crystals and quasicrystals.) The word "essentially" will be the difference between different mathematical definitions, which one derives from [1] and [2].

A (mathematical) quasicrystal is a tiling $T$ of ${\mathbb R}^n$ by convex polytopes satisfying certain properties:

1. Since general tilings are hard to work with, pretty much everybody assumes that $T$ is a Voronoi tiling of ${\mathbb R}^n$ based on a certain discrete subset $N\subset {\mathbb R}^n$, which a geometer would call a separated net.

2. Unfortunately, just having a separated net is not enough in order to deal with the "diffraction" issue. There is a disagreement which tilings are allowed as crystals and which are not. There is a general agreement that at least all periodic tilings (the ones where $N$ is a finite union of orbits of a discrete group of translations) and certain tilings constructed by Penrose and others via projection method, should be counted as crystals and quasicrystals respectively. I will call these two classes of tilings as "standard."

a. Here is Hof's definition of a crystal (Senchal's definition is taken from Hof's paper). Hof in [1] takes the auto-correlation function (actually, a distribution) $\gamma$ of $N$ and computes the (appropriately defined) Fourier transform $\hat\gamma$ of $\gamma$ (this is a mathematical interpretation of the "diffraction diagram"). Then $\hat\gamma$ is a measure $\mu$ which, in general, splits as a sum of two measures $\mu_d+\mu_c$: Discrete part $\mu_d$, which is supported on a certain countable subset of ${\mathbb R}^n$ and continuous part $\mu_c$. He proposes that "essentially discrete diffraction" means that $\mu_d$ is nonzero. Hof then proves that "standard" tilings indeed have nontrivial $\mu_d$ (According to [2], Hof even proves that $\mu_c=0$ in this case, but I did not check this). The trouble with this definition is that, as far as I can tell, there is no known purely geometric interpretation of the condition $\mu_d\ne 0$ in terms of the next $N$ itself (at least, none existed 6 years ago).

b. Gouere [2] (his work is an extension of Hof's approach and of a work by Lagarias) works with a slightly different definition, i.e., that $\mu_c=0$ (such sets $N$ are called Patterson sets). His main result is a purely geometric interpretation (actually, several interpretations) of this condition, see Theorem 1.1 in [1]: Patterson sets are the sets which are almost periodic with respect to Besikovitch's metric.

Remark 1. As far as I can tell from reading Freeman Dyson's paper here, definition of a quasicrystal that Dyson proposes is the one with $\mu_c=0$.

Remark 2. Gouere does not propose that a Patterson set is the right definition of a crystal, this is just my take on his paper. Condition $\mu_c=0$ is more limited, but, in view of [2], is geometric and also covers "standard" examples, while the condition $\mu_d\ne 0$ is more general, but is nongeometric.