Editing Roth's theorem (section)

==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.