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
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
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
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.
Notesedit
^H. Davenport, Wolfgang M. Schmidt, "Approximation to real numbers by quadratic irrationals," Acta Arithmetica 13, (1967).
Referencesedit
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
davenport, schmidt, theorem, mathematics, specifically, area, diophantine, approximation, tells, well, certain, kind, real, number, approximated, another, kind, specifically, tells, that, good, approximation, irrational, numbers, that, quadratic, using, either. 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 1 Statement 2 Notes 3 References 4 External linksStatement editGiven a number a 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 a 2 y a z 0 displaystyle x alpha 2 y alpha z 0 nbsp If a is a quadratic irrational we can take x y and z to be the coefficients of its minimal polynomial If a is rational we will have x 0 With these integers uniquely determined for each such a we can define the height of a to be H a max x y z displaystyle H alpha max x y z nbsp The theorem then says that for any real number 3 which is neither rational nor a quadratic irrational we can find infinitely many real numbers a which are rational or quadratic irrationals and which satisfy 3 a lt C H a 3 max 1 3 2 displaystyle xi alpha lt CH alpha 3 max 1 xi 2 nbsp where C is any real number satisfying C gt 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 3 Notes edit H Davenport Wolfgang M Schmidt Approximation to real numbers by quadratic irrationals Acta Arithmetica 13 1967 References editWolfgang 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 2000External links edit Davenport Schmidt theorem PlanetMath Retrieved from https en wikipedia org w index php title Davenport Schmidt theorem amp oldid 1124213288, wikipedia, wiki, book, books, library,
