Inhomogeneous approximation with coprime integers and lattice orbits

TL;DR

This paper explores inhomogeneous approximation with coprime integers and connects it to the density of lattice orbits, providing new variants of Minkowski's classical theorem.

Contribution

It introduces variants of Minkowski's theorem that incorporate coprimality constraints and relates these to lattice orbit density exponents.

Findings

New variants of Minkowski's theorem for coprime solutions

Connection established between inhomogeneous approximation and lattice orbit densities

Results extend classical approximation theory with coprimality considerations

Abstract

We obtain variants of the classical Minkowski Theorem on inhomogeneous approximation where we require moreover that the solutions $p, q$ be coprime integers. We link the subject with density exponents of lattice orbits in the real plane.