manuscripta mathematica

, Volume 94, Issue 1, pp 501–529 | Cite as

Metric results on the approximation of zero by linear combinations of independent and of dependent rationals

  • Gerhard Larcher
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Abstract

We give some “rational analoga” to metric results in the classical theory of the diophantine approximation of zero by linear forms. That is: we study the behaviour of expressions of the form
$$\begin{gathered} \lim _{m \to \infty } \frac{1}{{\left| {P_s (m)} \right|}}|\{ (x_1 , \ldots ,x_s ) \in P_s (m): \hfill \\ \parallel a_1 \frac{{x_1 }}{m} + \ldots + a_s \frac{{x_s }}{m}\parallel _m \geqslant \psi (a_1 , \ldots ,a_s ,m) \hfill \\ for all - \frac{m}{2}< a_1 , \ldots ,a_s \leqslant \frac{m}{2}, \hfill \\ with (a_1 , \ldots ,a_s ) \ne (0, \ldots ,0)\} |, \hfill \\ \end{gathered} $$
whereP s (m) is a certain subset of {1, …,m} s , ψ is a certain nonnegative function, and ‖ · ‖ m means the maximum of 1/m and the distance to the nearest integer. Some of the investigations are also motivated by problems in the theory of uniform distribution and of pseudo-random number generation. The results partly depend on the validity of the generalized Riemann hypothesis.

Keywords

Positive Integer Diophantine Approximation Quadratic Residue Dependent Quantity Legendre Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Gerhard Larcher
    • 1
  1. 1.Institut für MathematikUniversität SalzburgSalzburgAustria

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