# Equidistribution of divergent orbits and continued fraction expansion of rationals

@article{David2018EquidistributionOD,
title={Equidistribution of divergent orbits and continued fraction expansion of rationals},
author={Ofir David and Uri Shapira},
journal={Journal of the London Mathematical Society},
year={2018},
volume={98}
}
• Published 3 July 2017
• Mathematics
• Journal of the London Mathematical Society
We establish an equidistribution result for pushforwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis, we obtain new results regarding the asymptotic normality of the continued fraction expansions of most rationals with a high denominator as well as an estimate on the length of their continued fraction expansions.
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## References

SHOWING 1-10 OF 24 REFERENCES
Limiting distributions of translates of divergent diagonal orbits
• Mathematics
Compositio Mathematica
• 2019
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain
The distribution of closed geodesics on the modular surface, and Duke's theorem
• Mathematics
• 2011
We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who
Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids
• Mathematics
• 2014
For any non-uniform lattice Γ in SL2(ℝ), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL2(ℝ). As an application, for a quadratic form Q of signature (2,
On the number of solutions of the congruence ≡\pmod{} under the graph of a twice continuously differentiable function
. A result by V. A. Bykovski˘ı (1981) on the number of solutions of the congruence xy ≡ l (mod q ) under the graph of a twice continuously diﬀerentiable function is reﬁned. As an application,
ON THE NUMBER OF SOLUTIONS OF THE CONGRUENCE xy ≡ l (mod q) UNDER THE GRAPH OF A TWICE CONTINUOUSLY DIFFERENTIABLE FUNCTION
A result by V. A. Bykovskĭı (1981) on the number of solutions of the congruence xy ≡ l (mod q) under the graph of a twice continuously differentiable function is refined. As an application, Porter’s
Introduction to Ergodic theory
Hyperbolic dynamics studies the iteration of maps on sets with some type of Lipschitz structure used to measure distance. In a hyperbolic system, some directions are uniformly contracted and others
Closed orbits for actions of maximal tori on homogeneous spaces
• Mathematics
• 2003
Let G be a real algebraic group defined over Q, let 0 be an arithmetic subgroup, and let T be any torus containing a maximal R-split torus. We prove that the closed orbits for the action of T on G/0
Estimate for dispersion of lengths of continued fractions
An estimate for dispersion of lengths of continued fractions is proved for fixed denominator. This estimate improves the trivial one by the logarithm of the denominator.