Farey fractions and two-dimensional tori
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The Farey sequence gives a natural filtration with finite subsets of the set of rational numbers in [0, 1]. It is elementary to define this sequence and to prove that it is uniformly distributed on the interval [0, 1]. However, other aspects regarding its distribution appear to be quite intricate and are related with a number of important open problems in mathematics.
This survey paper revolves around three main themes. The first one is concerned with their spacing statistics, especially the existence and computation of consecutive level spacing measures. The second theme is the connection with some problems in geometric probability concerning the statistics of the linear flow in a punctured flat two-torus when the diameter of the puncture tends to zero. Finally some results and open problems about noncommutative two-tori, some of their subalgebras, and the spectral theory of almost Mathieu operators are reviewed. In particular, a connection between Farey fractions and the structure of gaps in the Hofstadter butterfly is discussed.
KeywordsChern Class Irrational Rotation Bratteli Diagram Harper Operator Farey Fraction
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