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.
Best Answer
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.
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.
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.