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| {{for|Roth's theorem on arithmetic progressions|Szemerédi's theorem}}
| | Biomedical Engineer Wesley Batterton from Mont-Saint-Hilaire, enjoys leathercrafting, property developers in singapore and swimming. In the last year has made a trip to Timbuktu.<br><br>My website ... [http://www.caseyreale.com/activity/p/227293/ http://www.caseyreale.com/activity/p/227293/] |
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| In [[mathematics]], the '''Thue–Siegel–Roth theorem''', also known simply as '''Roth's theorem''', is a foundational result in [[diophantine approximation]] to [[algebraic number]]s. It is of a qualitative type, stating that a given algebraic number <math>\alpha</math> may not have too many [[rational number]] approximations, that are 'very good'. Over half a century, the meaning of ''very good'' here was refined by a number of mathematicians, starting with [[Joseph Liouville]] in 1844 and continuing with work of {{harvs|txt|first=Axel|last= Thue|authorlink=Axel Thue|year=1909}}, {{harvs|txt|authorlink=Carl Ludwig Siegel|first=Carl Ludwig |last=Siegel|year=1921}}, {{harvs|txt|authorlink=Freeman Dyson|first=Freeman|last=Dyson|year=1947}}, and {{harvs|txt|authorlink=Klaus Roth|first=Klaus|last= Roth|year=1955}}.
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| == Statement ==
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| The Thue–Siegel–Roth theorem states that any [[Irrational number|irrational]] algebraic number <math>\alpha</math> has [[approximation exponent]] equal to 2, ''i.e.'', for given <math>\epsilon>0</math>, the inequality
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| :<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{2 + \epsilon}}</math>
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| can have only finitely many solutions in [[coprime integers]] <math>p</math> and <math>q</math>, as was conjectured by Siegel. Therefore any irrational algebraic α satisfies
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| :<math>\left|\alpha - \frac{p}{q}\right| > \frac{C(\alpha,\epsilon)}{q^{2 + \epsilon}}</math>
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| with <math>C(\alpha,\epsilon)</math> a positive number depending only on <math>\epsilon>0</math> and <math>\alpha</math>.
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| ==Discussion==
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| The first result in this direction is [[Liouville's theorem (transcendence theory)|Liouville's theorem]] on approximation of algebraic numbers, which gives an approximation exponent of ''d'' for an algebraic number α of degree ''d'' ≥ 2. This is already enough to demonstrate the existence of [[transcendental number]]s. Thue realised that an exponent less than ''d'' would have applications to the solution of [[Diophantine equation]]s and in '''Thue's theorem''' from 1909 established an exponent <math>d/2 + 1 + \epsilon</math>. Siegel's theorem improves this to an exponent about 2√''d'', and Dyson's theorem of 1947 has exponent about √(2''d'').
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| Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting ε = 0: by [[Dirichlet's theorem on diophantine approximation]] there are infinitely many solutions in this case. However, there is a stronger conjecture of [[Serge Lang]] that
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| :<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2 \log(q)^{1+\epsilon}}</math> | |
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| can have only finitely many solutions in integers ''p'' and ''q''. If one lets α run over the whole of the set of real numbers, not just the algebraic reals, then both Roth's conclusion and Lang's hold
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| for [[almost everywhere|almost all]] α. So both the theorem and the conjecture assert that a certain [[countable set]] misses a certain set of measure zero.
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| The theorem is not currently [[Effective results in number theory|effective]]: that is, there is no bound known on the possible values of ''p'',''q'' given α.<ref name=HindrySilverman344>{{cite book | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | authorlink2=Joseph H. Silverman | title=Diophantine Geometry: An Introduction | series=[[Graduate Texts in Mathematics]] | volume=201 | year=2000 | isbn=0-387-98981-1 | pages=344–345}}</ref> {{harvtxt|Davenport|Roth|1955}} showed that Roth's techniques could be used to give an effective bound for the number of ''p''/''q'' satisfying the inequality, using a "gap" principle.<ref name=HindrySilverman344/> The fact that we do not actually know ''C''(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.
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| == Proof technique ==
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| The proof technique was the construction of an [[auxiliary function]] in several variables, leading to a contradiction in the presence of too many good approximations. By its nature, it was ineffective (see [[effective results in number theory]]); this is of particular interest since a major application of this type of result is to bound the number of solutions of some [[diophantine equation]]s.
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| == Generalizations ==
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| There is a higher-dimensional version, [[subspace theorem|Schmidt's subspace theorem]], of the basic result. There are also numerous extensions, for example using the [[p-adic metric]],<ref>{{cite journal | first=D. | last=Ridout | title=The ''p''-adic generalization of the Thue-Siegel-Roth theorem | journal=[[Mathematika]] | volume=5 | pages=40–48 | year=1958 | zbl=0085.03501 }}</ref> based on the Roth method.
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| [[William J. LeVeque|LeVeque]] generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed [[algebraic number field]]. Define the ''[[height function|height]]'' ''H''(ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its [[Minimal polynomial (field theory)|minimal polynomial]]. Fix κ>2. For a given algebraic number α and algebraic number field ''K'', the equation
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| :<math> | \alpha - \xi | < \frac{1}{H(\xi)^\kappa} </math>
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| has only finitely many solutions in elements ξ of ''K''.<ref>{{cite book | last = LeVeque | first = William J. | authorlink = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 | origyear = 1956 | isbn = 978-0-486-42539-9 | zbl=1009.11001 | pages=II:148–152}}</ref>
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| ==See also==
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| *[[Davenport–Schmidt theorem]]
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| *[[Granville–Langevin conjecture]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{Citation | last1=Davenport | first1=H. | last2=Roth | first2=Klaus Friedrich | author1-link=Harold Davenport | author2-link=Klaus Friedrich Roth | title=Rational approximations to algebraic numbers | doi=10.1112/S0025579300000814 | mr=0077577 | zbl=0066.29302 | year=1955 | journal=[[Mathematika]] | issn=0025-5793 | volume=2 | pages=160–167}}
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| *{{Citation | last1=Dyson | first1=Freeman J. | author1-link=Freeman Dyson | title=The approximation to algebraic numbers by rationals | doi= 10.1007/BF02404697 | mr=0023854 | zbl=0030.02101 | year=1947 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=79 | pages=225–240}}
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| *{{Citation | last1=Roth | first1=Klaus Friedrich | author1-link=Klaus Friedrich Roth |title=Rational approximations to algebraic numbers | doi=10.1112/S0025579300000644 | mr=0072182 | year=1955 | journal=[[Mathematika]] | issn=0025-5793 | volume=2 | pages=1–20, 168 | zbl=0064.28501}}
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| * {{cite journal |author=[[Wolfgang M. Schmidt]] |title=Diophantine approximation |journal=[[Lecture Notes in Mathematics]] |volume=785 |publisher=Springer |year=1980, 1996 |doi=10.1007/978-3-540-38645-2}}
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| * {{cite journal |author=[[Wolfgang M. Schmidt]] |title=Diophantine approximations and Diophantine equations |journal=[[Lecture Notes in Mathematics]] |publisher=Springer Verlag |year=1991 |volume=1467 |doi=10.1007/BFb0098246}}
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| *{{Citation | last1=Siegel | first1=Carl Ludwig | author1-link=Carl Ludwig Siegel | title=Approximation algebraischer Zahlen | doi=10.1007/BF01211608 | year=1921 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=10 | issue=3 | pages=173–213}}
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| *{{Citation | last1=Thue | first1=A. | author1-link=Axel Thue | title=Über Annäherungswerte algebraischer Zahlen | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0135 | year=1909 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=135 | pages=284–305}}
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| ==Further reading==
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| * {{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 }}
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| * {{cite book | first1=Alan | last1=Baker | authorlink1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | authorlink2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 }}
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| * {{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 }}
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| * {{cite book | first=Paul | last=Vojta | authorlink=Paul Vojta | title=Diophantine Approximations and Value Distribution Theory | series=Lecture Notes in Mathematics | volume=1239 | publisher=[[Springer-Verlag]] | year=1987 | isbn=3-540-17551-2 | zbl=0609.14011 }}
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| {{DEFAULTSORT:Thue-Siegel-Roth Theorem}}
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| [[Category:Diophantine approximation]]
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| [[Category:Theorems in algebraic number theory]]
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Biomedical Engineer Wesley Batterton from Mont-Saint-Hilaire, enjoys leathercrafting, property developers in singapore and swimming. In the last year has made a trip to Timbuktu.
My website ... http://www.caseyreale.com/activity/p/227293/