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On a theorem of Davenport and Schmidt

Volume 198 / 2021

Nickolas Andersen, William Duke Acta Arithmetica 198 (2021), 37-75 MSC: 11J70, 11K60. DOI: 10.4064/aa191202-30-7 Published online: 10 December 2020


This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet’s theorems on the rational approximation of one or of two irrationals can be improved, and if so, by how much. We consider a generalization of this question in the simplest case of a single irrational but in the context of the geometry of numbers in $\mathbb R ^2$, with the sup-norm replaced by a more general one. Results include sharp bounds for how much improvement is possible under various conditions. The proofs use semiregular continued fractions that are characterized by a certain best approximation property determined by the norm.


  • Nickolas AndersenBYU Mathematics Department
    275 TMCB
    Provo, UT 84604, U.S.A.
  • William DukeUCLA Mathematics Department
    Box 951555
    Los Angeles, CA 90095-1555, U.S.A.

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