On the Lenstra constant associated to the Rosen continued fractions

TL;DR

This paper explores the relationship between the Legendre and Lenstra constants, demonstrating their equality under certain conditions, and provides explicit entropy values for Rosen continued fractions.

Contribution

It establishes the equivalence of Legendre and Lenstra constants when the Legendre constant exists, specifically for Rosen and alpha-continued fractions, and computes the Rosen map's entropy.

Findings

Legendre and Lenstra constants are equal when the Legendre constant exists

Explicit entropy value for the Rosen map with respect to its invariant measure

Applicability to Rosen and alpha-continued fractions

Abstract

The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and $\alpha$-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure.