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Davenport–Schmidt theorem

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How well a certain kind of real number can be approximated by another kind

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 α 2 + y α + z = 0. {\displaystyle 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 ( α ) = max { | x | , | y | , | z | } . {\displaystyle 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

| ξ − α | < C H ( α ) − 3 max ( 1 , ξ 2 ) , {\displaystyle |\xi -\alpha |<CH(\alpha )^{-3}\max(1,\xi ^{2}),}

where C is any real number satisfying C > 160/9.

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 )
  • Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000

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