Properties of the nearest integer continued fraction expansions

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 ${\tau }_{1\mathrm{/}2}$ which is absolutely continuous with respect to the Lebesgue measure.