Multiparametric geometry of numbers and its application to splitting transference theorems

TL;DR

This paper extends the parametric geometry of numbers to multiple parameters, applying it to weighted Diophantine approximation and lattice exponents, leading to refined transference theorems with chains of inequalities.

Contribution

It introduces a multiparametric framework for geometry of numbers and derives new inequalities for transference theorems in Diophantine approximation and lattice exponents.

Findings

Splits Dyson's transference theorem into chains of inequalities.

Provides a multiparametric approach to define intermediate exponents.

Derives an analogue of Khintchine's transference theorem for lattices.

Abstract

In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted Diophantine approximation, the second one concerns Diophantine exponents of lattices. In both settings we use multiparametric approach to define intermediate exponents. Then we split the weighted version of Dyson's transference theorem and an analogue of Khintchine's transference theorem for Diophantine exponents of lattices into chains of inequalities between the intermediate exponents we define basing on the intuition provided by the parametric approach.