Editing Roth's theorem

{{Short description|Algebraic numbers are not near many rationals}}
{{for|Roth's theorem on arithmetic progressions|Roth's theorem on arithmetic progressions}}
In [[mathematics]], '''Roth's theorem''' or '''Thue–Siegel–Roth theorem''' is a fundamental result in [[diophantine approximation]] to [[algebraic number]]s. It is of a qualitative type, stating that algebraic numbers cannot have many [[rational number|rational]] 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}}.

== Statement ==

Roth's theorem states that every [[Irrational number|irrational]] algebraic number <math>\alpha</math> has [[approximation exponent]] equal to 2. This means that, for every <math>\varepsilon>0</math>, the [[inequality (mathematics)|inequality]]

:<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{2 + \varepsilon}}</math>

can have only [[finite set|finitely many]] solutions in [[coprime]] [[integer]]s <math>p</math> and <math>q</math>. Roth's [[mathematical proof|proof]] of this fact resolved a [[conjecture]] by Siegel. It follows that every irrational algebraic number α satisfies

:<math>\left|\alpha - \frac{p}{q}\right| > \frac{C(\alpha,\varepsilon)}{q^{2 + \varepsilon}}</math>

with <math>C(\alpha,\varepsilon)</math> a positive number depending only on <math>\varepsilon>0</math> and <math>\alpha</math>.

==Discussion==

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''&nbsp;≥&nbsp;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 + \varepsilon</math> which he applied to prove the finiteness of the solutions of [[Thue equation]].  Siegel's theorem improves this to an exponent about 2{{radic|''d''}}, and Dyson's theorem of 1947 has exponent about {{radic|2''d''}}.

Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting <math>\varepsilon = 0</math>: 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

:<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2 \log(q)^{1+\varepsilon}}</math>

can have only finitely many solutions in integers ''p'' and ''q''.  If one lets α run over the whole of the set of [[real number]]s, not just the algebraic reals, then both Roth's conclusion and Lang's  hold
for [[almost everywhere|almost all]] <math>\alpha</math>. So both the theorem and the conjecture assert that a certain [[countable set]] misses a certain set of [[measure zero]].<ref>It is also closely related to the [[Arithmetic of abelian varieties#Manin–Mumford conjecture|Manin–Mumford conjecture]].</ref>

The theorem is not currently [[Effective results in number theory|effective]]: that is, there is no [[upper and lower bounds|bound]] known on the possible values of ''p'',''q'' given <math>\alpha</math>.<ref name=HindrySilverman344>{{citation | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | author-link2=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.

== Proof technique ==

The proof technique involves constructing an [[auxiliary function|auxiliary]] multivariate [[polynomial]] in an arbitrarily large number of variables depending upon <math>\varepsilon</math>, leading to a [[proof by contradiction|contradiction]] in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). 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 equations.

== Generalizations ==

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|''p''-adic metric]],<ref>{{citation | 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 | doi=10.1112/s0025579300001339}}</ref> based on the Roth method.

[[William J. 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 value]]s of the [[coefficient]]s of its [[minimal polynomial (field theory)|minimal polynomial]].  Fix κ>2.  For a given algebraic number α and algebraic number field ''K'', the equation

:<math>|\alpha - \xi| < \frac{1}{H(\xi)^\kappa}</math>

has only finitely many solutions in elements ξ of ''K''.<ref>{{citation | last = LeVeque | first = William J. | author-link = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 | orig-year = 1956 | isbn = 978-0-486-42539-9 | zbl = 1009.11001 | pages = [https://archive.org/details/topicsinnumberth0000leve/page/ II:148–152] | url = https://archive.org/details/topicsinnumberth0000leve/page/ }}</ref>

==See also==
*[[Davenport–Schmidt theorem]]
*[[Granville–Langevin conjecture]]
*[[Størmer's theorem]]
*[[Diophantine geometry]]

==Notes==
{{Reflist}}

==References==
*{{Citation | last1=Davenport | first1=H. | last2=Roth | first2=Klaus Friedrich | author1-link=Harold Davenport | author2-link=Klaus 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 | issue=2 | pages=160–167}}
*{{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| doi-access=free }}
*{{Citation | last1=Roth | first1=Klaus Friedrich | author1-link=Klaus 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}}
* {{citation |author=Wolfgang M. Schmidt |author-link=Wolfgang M. Schmidt |title=Diophantine Approximation |series=[[Lecture Notes in Mathematics]] |volume=785 |publisher=Springer |orig-year=1980|year=1996 |doi=10.1007/978-3-540-38645-2|isbn=978-3-540-09762-4 }}
* {{citation |author=Wolfgang M. Schmidt |author-link=Wolfgang M. Schmidt |title=Diophantine Approximations and Diophantine Equations |series=[[Lecture Notes in Mathematics]] |publisher=Springer-Verlag |year=1991 |volume=1467 |doi=10.1007/BFb0098246|isbn=978-3-540-54058-8 |s2cid=118143570 }}
*{{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 | mr=1544471| s2cid=119577458 | url=https://zenodo.org/record/1538156 }}
*{{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=1909 | issue=135 | pages=284–305 | doi=10.1515/crll.1909.135.284| s2cid=125903243 }}

==Further reading==
* {{citation | first=Alan | last=Baker | author-link=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 }}
* {{citation | first1=Alan | last1=Baker | author-link1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | author-link2=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 }}
* {{citation | first1=Enrico | last1=Bombieri | author-link1=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 }}
* {{citation | first=Paul | last=Vojta | author-link=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 }}

{{DEFAULTSORT:Thue-Siegel-Roth Theorem}}
[[Category:Diophantine approximation]]
[[Category:Theorems in number theory]]
[[Category:Algebraic numbers]]