About the value of the two dimensional Levy's constant

TL;DR

This paper numerically approximates the two-dimensional Levy's constant related to Diophantine approximations, expressing it as a 7-dimensional integral reduced to a triple integral for computational feasibility.

Contribution

It provides a numerical approximation of the 2D Levy's constant by reducing a 7D integral to a triple integral, enabling practical computation.

Findings

Numerical approximation of the 2D Levy's constant achieved.

Reduction of a 7D integral to a triple integral for computation.

Implementation of numerical methods for integral evaluation.

Abstract

We give a numerical approximation of the L\'evy constant on the growth of the denominators of the best Diophantine approximations in dimension 2 with respect to the euclidean norm. This constant is expressed as an integral on a surface of dimension 7. We reduce the computation of this integral to a triple integral, whose numerical evaluation was carried out in \cite{Xieu}.