Davenport–Schmidt theorem


Davenport–Schmidt theorem

In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. It is named after Harold Davenport and Wolfgang M. Schmidt.

Contents

Statement

Given a number α which is either rational or a quadratic irrational, we can find unique integers x, y, and z such that x, y, and z are not all zero, the first non-zero one among them is positive, they are relatively prime, and we have

x\alpha^2 +y\alpha +z=0.\,

If α is a quadratic irrational we can take x, y, and z to be the coefficients of its minimal polynomial. If α is rational we will have x = 0. With these integers uniquely determined for each such α we can define the height of α to be

H(\alpha)=\max\{|x|,|y|,|z|\}.\,

The theorem then says that for any real number ξ which is neither rational nor a quadratic irrational, we can find infinitely many real numbers α which are rational or quadratic irrationals and which satisfy

|\xi-\alpha|<CH(\alpha)^{-3},\,

where

C=\left\{\begin{array}{ c l } C_0 & \textrm{if}\ |\xi|<1 \\ C_0\xi^2 & \textrm{if}\ |\xi|>1.\end{array}\right.

Here we can take C0 to be any real number satisfying C0 > 160/9.[1]

While the theorem is related to Roth's theorem, its real use lies in the fact that it is effective, in the sense that the constant C can be worked out for any given ξ.

Notes

  1. ^ H. Davenport, Wolfgang M. Schmidt, "Approximation to real numbers by quadratic irrationals," Acta Arithmetica 13, (1967).

References

  • Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
  • Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000

External links


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