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.

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|

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