Unconventional height functions in simultaneous Diophantine approximation

Lior Fishman, David Simmons*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)577-618
Number of pages42
JournalMonatshefte fur Mathematik
Issue number3
Early online date18 Oct 2016
Publication statusPublished - 1 Mar 2017

Bibliographical note

© The Author(s) 2016.


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

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