Properties of the nearest integer continued fraction expansions

TL;DR

This paper investigates the properties of nearest integer continued fraction expansions, focusing on their metrical characteristics, approximation coefficients, and the stationary state of a related transformation, contributing to the understanding of these expansions.

Contribution

It introduces a new analysis of the metrical properties and stationary states of the nearest integer continued fraction expansions, including approximation coefficients and transformation dynamics.

Findings

Established metrical properties of the expansions

Defined and analyzed approximation coefficients

Proved existence of a stationary state for the transformation

Abstract

The nearest integer continued fraction of a real number $x$ from $[-1/2, 1/2)$ is defined. Some metrical properties of these expansions are presented. We define the approximation coefficients and give an important result on them. The main result consists in obtaining a stationary state for the transformation $\tau_{1/2}$ which is absolutely continuous with respect to the Lebesgue measure.