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Showing 1 to 7 of 7 for “"Farey fractions"”.
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Parabolas infiltrating the Ford circles
After briefly discussing classical results of Farey fractions and Ford circles, we define and study a new family of parabolas in connection with Ford circles and discover some interesting properties that they have. Using these properties, we provide an application of such a family.
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Visibility of lattice points in high dimensional spaces and extreme values of combinations of Dirichlet L-functions
… we study the phenomenon of similar ordering of Farey fractions and their generalizations. The notion of similar ordering for pairs of rationals was first introduced by Hardy, Littlewood and P\'olya. Later A.E. Mayer proved that pairs of Farey fractions in $\mathcal{F}_Q$ are similarly ordered …
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Transversals to horocycle flow on the moduli space of translation surfaces
… of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation …
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Correlations of sequences modulo one and statistics of geometrical objects associated to visible points
… for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical …
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Applications of dynamical systems to Farey sequences and continued fractions
… in SL(2,R) to study the spacing statistics of Farey fractions. For a given finite index subgroup H ⊆ SL(2,Z), we use a process developed by Fisher and Schmidt to lift a cross section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of …
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Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations
K.T. Atanassov introduced the two arithmetic functions \[ I(n) = \prod_{\nu=1}^k p_\nu^{1/\alpha_\nu} \qquad \text{and}\qquad R(n) = \prod_{\nu=1}^k p_\nu^{\alpha_v - 1} \] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties …
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Beatty ratios, algorithms related to Sturmian sequences and uniform distribution theory
… use Diophantine approximation, continued fractions and various number theoretic and combinatorial arguments. The third portion of this thesis deals with a topic related to a theorem that begun as a conjecture of Steinhaus. This well known Three Gap Theorem states that there are at most …