Geometry of Diophantine exponents

TL;DR

This survey reviews the current understanding of Diophantine exponents, their relations, and various forms, highlighting the importance of the transference principle in approximation theory.

Contribution

It provides a comprehensive overview of Diophantine exponents, including classical, weighted, multiplicative, and lattice cases, emphasizing recent developments and open problems.

Findings

Summary of classical Diophantine exponents and their properties.

Discussion of Diophantine exponents with weights and multiplicative variants.

Analysis of the transference principle and its implications.

Abstract

Diophantine exponents are ones of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation which studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle.