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Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?

I've heard Freeman Dyson say that the zeros of the Riemann zeta function form a quasi-crystal. But, a priori, I do not see what kind of property of the zeros, that we currently now of, would be able to confer to them more structure than to a random set of isolated numbers.

(Notwithstanding the explicit formula in prime number theory)

To wit, my second question possibly based on a misunderstanding: why is the set of zeros of $\zeta(s)$ a quasi-crystal, while a random sequence of isolated numbers is not? Of course, I first need to fully understand what is a quasi-crystal, because Freeman's definition left me in a fog.

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  • $\begingroup$ Inquiring minds want to know. $\endgroup$
    – kolik
    May 28, 2012 at 7:08
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    $\begingroup$ What makes you think Pisot numbers relate to quasi-crystals and/or the Riemann Hypothesis? Did Dyson say something about those, too? $\endgroup$
    – Gerry Myerson
    May 28, 2012 at 7:21
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    $\begingroup$ en.wikipedia.org/wiki/Quasicrystal. Please try Wikipedia before posting here. $\endgroup$
    – Charles Matthews
    May 28, 2012 at 10:37
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    $\begingroup$ Dyson's definition of a quasicrystal is not equivalent to the one in Wikipedia. $\endgroup$
    – Misha
    May 28, 2012 at 13:35
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    $\begingroup$ @Charles: The author of wikipedia article is not a mathematician, so he/she does not understand the difference between the words "define" and "construct." $\endgroup$
    – Misha
    May 28, 2012 at 19:01

1 Answer 1

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Freeman Dyson's proposal is online, based on a talk he gave at MSRI.

Lillian Pierce's senior thesis gives a summary of Peter Sarnak's program to use properties of Gaussian Unitary Ensemble to study the zeros of the Riemann Zeta function.

N. G. Debrujin wrote about Penrose tilings and their Fourier transforms.

Crystalline structures on the line are pretty boring. They are just evenly spaced lattices, like $\mathbb{Z}$, which might appear on different scales.

--o---o---o---o---o---o---o--
---o-----o-----o-----o-----o-

However, there are many quasi-periodic structures on the line, for example $\lfloor n\sqrt{2}\rfloor = \{ 1, 2, 4, 5, 7, 8, 9, 11, 12, 14,\dots \}$ which we can draw on the line.

--o--o-----o--o-----o--o--o-----o--o-----o--

Many of these have special recursive properties. Consider the line $y = \frac{1 + \sqrt{5}}{2} x$ which Golden ration slope. Mark "0" if it crosses a horizontal line and "1" if for a vertical line. You get the Fibonacci Word Of course in 2D you get more interesting quasicrystals, which have interesting number theoretic and recursive structures.
Freeman Dyson wishes the zeros of the Riemann Hypotheses have structure like these.

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    $\begingroup$ @John: What your definition of quasi-periodicity? As far as I understand the question, one issue is lack of precise definitions in the quasicrystal literature, which is dominated by physics papers. Also, your 2nd link is broken. $\endgroup$
    – Misha
    May 28, 2012 at 18:37
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    $\begingroup$ See the newer question: mathoverflow.net/questions/133581/… $\endgroup$
    – user25199
    Jun 13, 2013 at 13:56

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