Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion

TL;DR

This paper studies the distribution of special quadratic irrationals from the odd continued fraction expansion, showing they are equidistributed according to a specific invariant measure on a modular surface.

Contribution

It introduces the concept of O-reduced quadratic irrationals and proves their equidistribution with respect to the Lebesgue measure for the odd Gauss shift.

Findings

O-reduced quadratic irrationals are equidistributed

Distribution aligns with the invariant measure of the odd Gauss shift

Connects geodesic length to distribution of irrationals

Abstract

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface $\Gamma \backslash \mathbb{H}$, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.