×

A transfer property in Diophantine approximation. (Une propriété de transfert en approximation diophantienne.) (French) Zbl 1065.37005

The author investigates the well-known link between the properties of uniform simultaneous rational approximations for a vector \(w\) in \(\mathbb{R}^n\) and the sequence of best simultaneous rational approximations for such a vector \(w\). The study makes use of a dynamical framework. In particular, the quality of the best simultaneous rational approximations to the vector \(w\) is expressed in terms of the best return times near the origin for the translation \(x\mapsto x+ w\) on the \(n\)-dimensional torus.

MSC:

37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11J13 Simultaneous homogeneous approximation, linear forms
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Jacobson, N. , Basic Algebra I, second edition, W. H. Freeman and Company , New York, (1985). · Zbl 0557.16001
[2] Lochak, P., Canonical perturbation theory via simultaneous approximation, Russ. Math. Surveys47, p. 57-133 (1992). · Zbl 0795.58042
[3] Schmidt, W.M. , Diophantine Approximation, , 785 (1980). · Zbl 0421.10019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.