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
- [math]\displaystyle{ x\alpha^2 +y\alpha +z=0. }[/math]
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
- [math]\displaystyle{ H(\alpha)=\max\{|x|,|y|,|z|\}. }[/math]
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
- [math]\displaystyle{ |\xi-\alpha|\lt CH(\alpha)^{-3}\max(1,\xi^2), }[/math]
where C is any real number satisfying C > 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
- ↑ 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
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