Multidimensional statistics for finite orbits of generalised continued fractions

2025/07/15 by Berthé, Valérie, Cantrell, Stephen, Lee, Jungwon +1
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.11295

Abstract

We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and Central Limit Theorem, including multidimensional results for random vectors. These results apply to classical multidimensional continued fraction transformations including Brun's algorithm and the Jacobi--Perron algorithm, and more generally for maps satisfying mild contraction hypothesis on the inverse branches. We prove in particular that the finite trajectories capture the generic ergodic behaviour of infinite trajectories.

Citations

• Euclidean algorithms are Gaussian over imaginary quadratic fields
• Limit laws for rational continued fractions and value distribution of quantum modular forms
• Dynamics of continued fractions and distribution of modular symbols
• A short proof that the number of division steps in the Euclidean algorithm is normally distributed
• The dynamics of Pythagorean triples
• Euclidean algorithms are Gaussian

Related

• Rational approximations, multidimensional continued fractions and lattice reduction
• On a continued fraction algorithm in finite extensions of \Q_p and its metrical theory
• On the rate of convergence of continued fraction statistics of random rationals
• Invariant measures for continued fraction algorithms with finitely many digits
• Continued fractions, the Chen-Stein method and extreme value theory