Editing Diophantine approximation (section)

=== Approximation of algebraic numbers, Thue–Siegel–Roth theorem ===
{{main|Thue–Siegel–Roth theorem}}

Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to {{harvs|first=Axel|last=Thue|authorlink=Axel Thue|year=1909|txt}}, {{harvs|frst=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1921|txt}}, {{harvs|first=Freeman|last=Dyson|authorlink=Freeman Dyson|year=1947|txt}}, and {{harvs|first=Klaus|last=Roth|authorlink=Klaus Roth|year=1955|txt}}, leading finally to the Thue–Siegel–Roth theorem: If {{math|''x''}} is an irrational algebraic number and {{math|''ε > 0''}}, then there exists a positive real number {{math|''c''(''x'', ''ε'')}} such that
:<math>
    \left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}
</math>
holds for every integer {{math|''p''}} and {{math|''q''}} such that {{math|''q'' > 0}}.

In some sense, this result is optimal, as the theorem would be false with ''ε''&nbsp;=&nbsp;0. This is an immediate consequence of the upper bounds described below.