Unconventional height functions in simultaneous Diophantine approximation

Lior Fishman; David Simmons

doi:10.1007/s00605-016-0983-0

Unconventional height functions in simultaneous Diophantine approximation

Lior Fishman, David Simmons^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Simultaneous Diophantine approximation is concerned with the approximation of a point x∈ R^d by points r∈ Q^d, with a view towards jointly minimizing the quantities ‖ x- r‖ and H(r). Here H(r) is the so-called “standard height” of the rational point r. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.

Original language	English
Pages (from-to)	577-618
Number of pages	42
Journal	Monatshefte fur Mathematik
Volume	182
Issue number	3
Early online date	18 Oct 2016
DOIs	https://doi.org/10.1007/s00605-016-0983-0
Publication status	Published - 1 Mar 2017

Bibliographical note

Keywords

Continued fractions
Diophantine approximation
Dirichlet’s theorem
Hardy L-functions
Height functions

Access to Document

10.1007/s00605-016-0983-0

1401.8266Accepted author manuscript, 324 KB
art_10.1007_s00605-016-0983-0
© The Author(s) 2016.
Final published version, 532 KBLicence: CC BY

https://arxiv.org/abs/1401.8266

Cite this

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abstract = "Simultaneous Diophantine approximation is concerned with the approximation of a point x∈ Rd by points r∈ Qd, with a view towards jointly minimizing the quantities ‖ x- r‖ and H(r). Here H(r) is the so-called “standard height” of the rational point r. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.",

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KW - Diophantine approximation

KW - Dirichlet’s theorem

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Unconventional height functions in simultaneous Diophantine approximation

Abstract

Bibliographical note

Keywords

Access to Document

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Programme Grant-New Frameworks in metric Number Theory

Cite this

Unconventional height functions in simultaneous Diophantine approximation

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Projects

Programme Grant-New Frameworks in metric Number Theory

Cite this