Vol. 11, No. 1, 2022

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On the approximation exponents for subspaces of $\mathbb{R}^n$

Elio Joseph

Vol. 11 (2022), No. 1, 21–35
Abstract

This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of n established by W. M. Schmidt in 1967. Let A and B be two subspaces of n of respective dimensions d and e with d + e n. The proximity between A and B is measured by t = min (d,e) canonical angles 0 𝜃1 𝜃t π 2 ; we set ψj(A,B) = sin 𝜃j. If B is a rational subspace, its complexity is measured by its height H(B) = covol (B n). We denote by μn(A|e)j the exponent of approximation defined as the upper bound (possibly equal to + ) of the set of β > 0 such that the inequality ψj(A,B) H(B)β holds for infinitely many rational subspaces B of dimension e. We are interested in the minimal value μ¨n(d|e)j taken by μn(A|e)j when A ranges through the set of subspaces of dimension d of n such that for all rational subspaces B of dimension e one has dim (A B) < j. We show that μ¨4(2|2)1 = 3, μ¨5(3|2)1 6 and μ¨2d(d|)1 2d2(2d ). We also prove a lower bound in the general case, which implies that μ¨n(d|d)d 1d as n +.

Keywords
Diophantine approximation, rational subspaces, number theory, approximation exponents
Mathematical Subject Classification
Primary: 11J13
Secondary: 11J25
Milestones
Received: 8 June 2021
Revised: 31 December 2021
Accepted: 20 January 2022
Published: 30 March 2022
Authors
Elio Joseph
Université Paris-Saclay
CNRS, Laboratoire de mathématiques d’Orsay
Orsay
France