On the distribution of partial quotients of reduced fractions with fixed denominator

TL;DR

This paper investigates the distribution of partial quotients in continued fractions of fractions with a fixed denominator, revealing their close alignment with the Gauss-Kuzmin distribution and establishing new concentration and tail estimates.

Contribution

It provides new concentration inequalities, tail estimates, and demonstrates the distribution's similarity to the Gauss-Kuzmin law for fractions with a fixed denominator.

Findings

Distribution closely matches Gauss-Kuzmin distribution

Sharp tail estimates for maximal partial quotient and Dedekind sums

Existence of fractions with small sum and maximal partial quotients

Abstract

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with $N$. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators $N$. We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauss-Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient.