Exponents of inhomogeneous Diophantine Approximation

TL;DR

This paper introduces new exponents in inhomogeneous Diophantine approximation, establishing a fundamental relation between approximation exponents of generic points and dual forms, thereby enhancing understanding of classical transference principles.

Contribution

It defines new Diophantine exponents for inhomogeneous problems and proves a key equality linking these to homogeneous exponents via dual forms.

Findings

Exponent of approximation to a generic point equals the inverse of the homogeneous exponent of dual forms.

Revisits classical transference theorems with new exponents.

Establishes a fundamental relation between inhomogeneous and homogeneous exponents.

Abstract

In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.