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SchÄffer’s Determinant Argument

  • Roger C. Baker
Conference paper
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

Let ‖...‖ denote distance from the nearest integer. Various versions of the following problem in simultaneous Diophantine approximation have been studied since 1957, beginning with Danicic [5]. Given an integer h ≥ 2. we seek a number θ having the following property, for every ∈ > 0 and every pair α = (α1, ... αh), β = (β1,..., βh) in ℝh: For N > C(h, ∈, there is an integer n, 1 ≤ nN, satisfying
$$ \left\| {n^2 \alpha j + n\beta j} \right\| < N^{ - \theta + \in } \left( {j = 1, \ldots ,h} \right). $$

Keywords

Fractional part quadratic polynomial exponential sum lattice successive minima 

2000 Mathematics subject classification

11J54 

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References

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    Danicic, I.: Contributions to number theory, Ph.D. thesis, University of London, London, United Kingdom (1957)Google Scholar
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    Schmidt, W.M.: Diophantine Approximation and Diophantine Equations. Lect. Notes Math., vol. 1467. Springer, Heidelberg (1991)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Roger C. Baker
    • 1
  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA

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