Bellissard's gap labeling conjecture for quasicrystals without magnetic field
Let be the quasicrystal with transversal , and let be its twisted groupoid -algebra for zero magnetic field. The trace induces the gap-labeling map
The set of gap labels is given by
which is precisely the group generated by the patch frequencies of . This conjecture predicts that, in the absence of a magnetic field, all gap labels are determined by the transversal and its patch frequencies. The source does not state whether the conjecture has been resolved.
References
Primary source
Michael Kreisel, “Gabor Frames for Quasicrystals, K-theory, and Twisted Gap Labeling”, arXiv:1411.7269 (2014).
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