Equidistribution of continued fraction convergents in $\mathrm{SL}(2,\mathbb{Z}_m)$ with an application to local discrepancy

TL;DR

The paper studies the equidistribution of continued fraction convergents modulo $m$ by embedding the natural recursions into the group $\mathrm{SL}(2,\mathbb{Z}_m)$ via the matrix sequence $P_n$. It proves central limit, law of the iterated logarithm, and invariance principles for functions of $P_n$, using a Gauss--Kuzmin--Lévy framework for a skew product and establishing exponential mixing. As an application, it derives limit laws for the local discrepancy of irrational rotations, including explicit normalizing constants in Kesten-type laws and joint stable-limit results for maximal Birkhoff sums, thereby linking Diophantine approximation modulo $m$ with detailed probabilistic limit theorems. The work also provides a thorough group-theoretic treatment of $\mathrm{SL}(2,\mathbb{Z}_m)$ and clarifies the dependence of constants on the test interval and modulus. Overall, it advances probabilistic limit theory for continued fractions in finite groups and yields precise local-discrepancy statistics with explicit constants.

Abstract

Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of the $2 \times 2$ matrix with entries $p_{n-1}, p_n, q_{n-1}, q_n \pmod{m}$. Improving the strong law of large numbers due to Szüsz, Moeckel, Jager and Liardet, we establish the central limit theorem and the law of the iterated logarithm, as well as the weak and the almost sure invariance principles. As an application, we find the limit distribution of the maximum and the minimum of the Birkhoff sum for the irrational rotation with the indicator of an interval as test function. We also compute the normalizing constant in a classical limit law for the same Birkhoff sum due to Kesten, and dispel a misconception about its dependence on the test interval.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Remark
• proof : Proof of Theorem \ref{['kestentheorem']}
• Lemma 6