# Distribution of approximants and geodesic flows

@article{Fisher2013DistributionOA, title={Distribution of approximants and geodesic flows}, author={Albert M. Fisher and Thomas A. Schmidt}, journal={Ergodic Theory and Dynamical Systems}, year={2013}, volume={34}, pages={1832 - 1848} }

Abstract We give a new proof of Moeckel’s result that for any finite index subgroup of the modular group, almost every real number has its regular continued fraction approximants equidistributed into the cusps of the subgroup according to the weighted cusp widths. Our proof uses a skew product over a cross-section for the geodesic flow on the modular surface. Our techniques show that the same result holds true for approximants found by Nakada’s $\alpha $-continued fractions, and also that the…

## 8 Citations

Gap distribution of Farey fractions determined by subgroups of SL(2,Z)

- Mathematics
- 2014

For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincare section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the…

Poincaré sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics

- Mathematics
- 2014

For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincar\'e section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to…

The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups $G_q$

- Mathematics
- 2018

The Farey sequence $\mathcal{F}(Q)$ at level $Q$ is the sequence of irreducible fractions in $[0, 1]$ with denominators not exceeding $Q$, arranged in increasing order of magnitude. A simple…

Non-trivial matrix actions preserve normality for continued fractions

- MathematicsCompositio Mathematica
- 2017

A seminal result due to Wall states that if $x$ is normal to a given base $b$ , then so is $rx+s$ for any rational numbers $r,s$ with $r\neq 0$ . We show that a stronger result is true for normality…

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$SL(2,R)/SL(2,Z) and applications to Farey fraction statistics

- Mathematics
- 2016

For a given finite index subgroup $$H\subseteq \mathrm {SL}(2,\mathbb {Z})$$H⊆SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on $$\mathrm…

Equidistribution of Farey sequences on horospheres in covers of SL(n+1,Z)\SL(n+1,R) and applications

- Mathematics
- 2017

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers…

Equidistribution of Farey sequences on horospheres in covers of and applications

- MathematicsErgodic Theory and Dynamical Systems
- 2019

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash…

Equidistribution of Farey sequences on horospheres in covers of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$ and applications

- Ergodic Theory and Dynamical Systems
- 2021

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash…

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