en>Jason Quinn: /* Results */ fixed deprecated month paramater in cite template

2013-12-20T06:01:36Z

Results: fixed deprecated month paramater in cite template

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			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 irrational]]s or simply [[rational number]]s. 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

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			If α is a quadratic irrational we can take ''x'', ''y'', and ''z'' to be the coefficients of its [[minimal polynomial (field theory)\|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>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>\|\xi-\alpha\|<CH(\alpha)^{-3},\,</math>

			where

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

			Here we can take ''C''<sub>0</sub> to be any real number satisfying ''C''<sub>0</sub> > 160/9.<ref>H. Davenport, Wolfgang M. Schmidt, "''Approximation to real numbers by quadratic irrationals''," Acta Arithmetica '''13''', (1967).</ref>

			While the theorem is related to [[Thue–Siegel–Roth theorem\|Roth's theorem]], its real use lies in the fact that it is [[Effective results in number theory\|effective]], in the sense that the constant ''C'' can be worked out for any given ξ.

			==Notes==
			<references/>

			==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==
			*{{planetmath reference\|id=4151\|title=Davenport-Schmidt theorem}}

			{{DEFAULTSORT:Davenport-Schmidt theorem}}
			[[Category:Diophantine approximation]]
			[[Category:Theorems in number theory]]

en>Chenxlee: /* The problem */ Typo

2011-05-25T11:03:31Z

The problem: Typo

New page

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Constant problem - Revision history

en>Jason Quinn: /* Results */ fixed deprecated month paramater in cite template

en>Chenxlee: /* The problem */ Typo