On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

TL;DR

This paper investigates the distribution of minimal solutions to linear Diophantine equations and demonstrates that the ratio of their Euclidean norms to coefficients is uniformly distributed modulo one, linking it to hyperbolic space dynamics.

Contribution

It establishes a connection between minimal solutions of Diophantine equations and the uniform distribution of orbits in hyperbolic spaces, extending previous work by Dinaburg and Sinai.

Findings

The signed ratio of minimal solutions' norms is uniformly distributed modulo one.

The problem reduces to an equidistribution theorem for Fuchsian group orbits.

Provides a new perspective on the statistical behavior of Diophantine solutions.

Abstract

We study a variant of a problem considered by Dinaburg and Sinai on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.