What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:39:04Z http://mathoverflow.net/feeds/question/98171 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers? kolik 2012-05-28T05:56:25Z 2012-05-28T17:22:39Z <p>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?</p> <p>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. </p> <p>(Notwithstanding the explicit formula in prime number theory)</p> <p>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. </p> http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers/98185#98185 Answer by John Mangual for What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers? John Mangual 2012-05-28T11:33:25Z 2012-05-28T17:22:39Z <p><a href="http://www.math.ucdavis.edu/~oyounggo/books/dyson.pdf" rel="nofollow">Freeman Dyson's proposal</a> is online, based on a talk he gave at <a href="http://www.msri.org/" rel="nofollow">MSRI</a>.</p> <p>Lillian Pierce's <a href="http://people.maths.ox.ac.uk/piercel/theses/Pierce_senior_Thesis.pdf" rel="nofollow">senior thesis</a> gives a summary of Peter Sarnak's program to use properties of Gaussian Unitary Ensemble to study the zeros of the Riemann Zeta function.</p> <p>N. G. Debrujin wrote about <a href="http://alexandria.tue.nl/repository/freearticles/597566.pdf" rel="nofollow">Penrose tilings</a> and their <a href="http://alexandria.tue.nl/repository/freearticles/597578.pdf" rel="nofollow">Fourier transforms</a>.</p> <hr> <p>Crystalline structures on the line are pretty boring. They are just evenly spaced lattices, like $\mathbb{Z}$, which might appear on different scales.</p> <pre>--o---o---o---o---o---o---o-- ---o-----o-----o-----o-----o-</pre> <p>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.</p> <pre>--o--o-----o--o-----o--o--o-----o--o-----o-- </pre> <p>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 <img src="http://upload.wikimedia.org/wikipedia/commons/2/2d/Fibonacci_word_cutting_sequence.png"/> Of course in 2D you get more interesting quasicrystals, which have interesting number theoretic and recursive structures.<br> <img src="http://upload.wikimedia.org/wikipedia/commons/1/1a/Penrose_Tiling_%28Rhombi%29.svg"/> Freeman Dyson wishes the zeros of the Riemann Hypotheses have structure like these.</p>