Mathematics – Dynamical Systems
Scientific paper
2012-01-05
Mathematics
Dynamical Systems
Scientific paper
We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' statistics given by the Gauss-Kuzmin measure. As far as we know, these are the first non-average results about the statistics of the periods of quadratic irrationals. As a by-product, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer $k$ and a quadratic irrational $\alpha$, the length of the period of the continued fraction expansion of $k^n\alpha$ equals $c' k^n +o(k^{(1-1/16)n})$ for some positive constant $c'$. This improves results of Lagarias and Grisel. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces.
Aka Menny
Shapira Uri
No associations
LandOfFree
On the evolution of continued fractions in a fixed quadratic field does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the evolution of continued fractions in a fixed quadratic field, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the evolution of continued fractions in a fixed quadratic field will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-661084