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{{for|Roth's theorem on arithmetic progressions|Szemerédi's theorem}}
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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}}.
 
== Statement ==
 
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
 
:<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{2 + \epsilon}}</math>
 
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
 
:<math>\left|\alpha - \frac{p}{q}\right| > \frac{C(\alpha,\epsilon)}{q^{2 + \epsilon}}</math>
 
with <math>C(\alpha,\epsilon)</math> a positive number depending only on <math>\epsilon>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 + \epsilon</math>. Siegel's theorem improves this to an exponent about 2√''d'', and Dyson's theorem of 1947 has exponent about √(2''d'').
 
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
 
:<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2 \log(q)^{1+\epsilon}}</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 numbers, not just the algebraic reals, then both Roth's conclusion and Lang's  hold
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.
 
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.
 
== Proof technique ==
 
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.
 
== 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]],<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.
 
[[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
 
:<math> | \alpha - \xi | < \frac{1}{H(\xi)^\kappa} </math>
 
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>
 
==See also==
*[[Davenport–Schmidt theorem]]
*[[Granville–Langevin conjecture]]
 
==Notes==
<references/>
 
==References==
*{{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}}
*{{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}}
*{{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}}
* {{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}}
* {{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}}
*{{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}}
*{{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}}
 
==Further reading==
* {{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 }}
* {{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 }}
* {{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 }}
* {{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 }}
 
{{DEFAULTSORT:Thue-Siegel-Roth Theorem}}
[[Category:Diophantine approximation]]
[[Category:Theorems in algebraic number theory]]

Revision as of 02:01, 26 February 2014

With greater than 30,000 property brokers in Singapore, turning into a Profitable property agent takes greater than only a license and information of present legal guidelines and laws. The primary 12 months drop out rate is estimated to be from 40% to eighty% shows that many actual estate brokers are usually not as profitable as they could be. Statistics suggests that 90% surrender after 3 years.

In Singapore Legal expenses are very high. Incase of an issue with any deal, if there's a legal challenge the true estate agent can seek the advice of the in-home firm lawyer totally free and sometimes if it is extremely complicated for a really subsidized amount. It's no surprise that real estate agents choose to work with large corporations. RB Capital Group set to rebrand the Gallery Resort in Singapore Diploma In Constructing and Property Management Newest Property Developments Property Brokers Information & Guides paperless submission of Sale or Rental transactions Services embrace letting, renting and property sales. Citiprop Property Administration Pte Ltd Particular person and corporate relocations for rental properties. 314 Tanglin Rd, Block F #01-01/02, Phoenix Park Office Campus, Singapore 247977. Industrial

Negotiation is a sophisticated ability that may contribute to the success of your sale. To do so, you have to suppress your emotions to forestall conflicts, deal with unreasonable criticisms, and respond to purchaser's whims and fancies. It also contains responding to snide remarks, answering frustrating questions, and negotiating terms and conditions in your favor. The subsequent step is a vital one. When you're an organization, partnership or sole-proprietor (not people) whose business involves shopping for, selling and leasing properties, then you definately'll want to apply for a House Agent's License. It's good to get this license from Singapore's IRAS (Inland Revenue Authority) Introduce and be certain that all actual estate brokers are registered. Genuine Wonderful Service Housing Allowances

There needs to be no causes for the authorities to reject your utility if you have submitted all the necessary documents and accurate information. As a way to save each your time and sources, you'll be able to rent an expert services firm to handle your House Agent's License application. Renewing your License Developers are actually trying to transfer the chance of an unknown market to the buyers. Nonetheless, many patrons, particularly first-time buyers and HDB upgraders, are excited by the new spherical of discounts and may't wait to hurry into the market. Upon expiry of the tenancy, help you within the negotiation for a contemporary lease or in the search for different lodging. Please call (sixty five) 9295 9180 or get in contact through the form on the top right of this web page.

We often hear tales of top property agents making an excellent revenue and stroll away with the impression that there is simple money to be made in the real property industry. This evaluation reveals that for every million-dollar property agent, there are many others who do not make it and finally fall out of the sport. Hence, it doesn't come as a shock that the real property business has such a excessive turnover charge. Perhaps because of this about three,seven hundred (greater than 10% of the entire agent inhabitants) decided not to renew their license and we're right down to 30,600 brokers in 2012.

Her experience greatly advantages her clients be it property sellers, buyers or tenants in search of an excellent place efficiently. More data for our German clients please go to the German/Deutsch language information. Over the years, OrangeTee has expanded into each side of real estate, and we've enabled our associates to direct their clients to optimum investment alternatives and help property homeowners to maximise the value of their assets. As the worldwide real estate market becomes extra subtle and international real property investments increases, the PERIOD actual property community is well geared up to supply professional recommendation and guidance to our purchasers in making crucial actual property decisions. Particular initiatives to enhance the value of the estate

Singapore Livestock Farmers Association Singapore Hong Lim Complicated Retailers' Association Singapore Mini Mart Association Singapore People's Park Centre Merchants' Affiliation Singapore Individuals's Park Advanced Merchants Association Singapore Provision Store Pleasant Association Singapore Religions Goods Merchants Association The Federation of Retailers' Associations, Singapore The Singapore Booksellers & Stationers Association Singapore Textile Dealers Friendly Affiliation Textile & Trend Federation (Singapore) Singapore Lighter House owners' Affiliation Singapore Motor Cycle Commerce Affiliation Spa & Wellness Affiliation (singapore hdb rental) Downtown Core, Singapore - $7-eight per hour Property Search Companies for investors and residential buyers Singapore Citizen or Everlasting Resident or Property Guides Property Highlights

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