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Multiparametric geometry of numbers and its application to splitting transference theorems

Multiparametric geometry of numbers and its application to splitting transference theorems 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 based on the intuition provided by the parametric approach. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte für Mathematik Springer Journals

Multiparametric geometry of numbers and its application to splitting transference theorems

Monatshefte für Mathematik , Volume 197 (4) – Apr 1, 2022

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021
ISSN
0026-9255
eISSN
1436-5081
DOI
10.1007/s00605-021-01651-4
Publisher site
See Article on Publisher Site

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 based on the intuition provided by the parametric approach.

Journal

Monatshefte für MathematikSpringer Journals

Published: Apr 1, 2022

Keywords: Parametric geometry of numbers; Diophantine approximation with weights; Lattice exponents; Transference principle; 11J13; 11H46; 11H60

References