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This is a question about the Farey sequences (Fn) in the limit case as n approaches infinity. If we take the infinite case of the Farey sequences and treat all the points in this set as a linear array, is this set of points a 1D quasicrystal?

MargaretWertheim
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  • I do not understand what you mean by "limit case". Farey numbers are used in hyperbolic geometry. Perhaps you are defining the boundary of the hyperbolic plane? Then you get a round circle. – markvs Jun 25 '20 at 04:57
  • I've enjoyed your books, Margaret, but I'm having trouble with your question. The "infinite case of the Farey sequences" is just the set of all rationals in the unit interval, no? – Gerry Myerson Jun 25 '20 at 05:54
  • Thanks Gerry. I understand this is the case. But can the rationals be regarded in any sense as a "quasicrystal"? It may be they can't. I'm interested because of Freeman Dyson's suggestion in a 2008 address to the AMS that one way we might prove the Riemann hypothesis is by finding a 1D quasicrystal that corresponds to the distribution of non-trivial zeroes along the X=1/2 line. Farey sequences have also been related to the Riemann hypothesis. Is it possible they represent a quasicrystal structure? I'm writing an article trying to explain this to a general audience and welcome any feedback. – MargaretWertheim Jun 26 '20 at 05:15
  • Sorry, I don't know enough about quasicrystals to tell when something is or isn't one. But aren't quasicrystals supposed to be discrete (which the rationals certainly are not)? By the way, if you want to be sure I see a comment intended for me, you have to have @Gerry in it. – Gerry Myerson Jun 26 '20 at 12:41
  • @Gerry, thanks for this. On thinking more about the problem I'm trying to describe, I'm sure now that Farey sequences aren't a quasicrystal. There 's another sequence allied with them that I'm interested in, and I'm trying to see if it may be a QC or not. I am inclined now to say Not. Thanks again – MargaretWertheim Jun 27 '20 at 19:11

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