Editing Diophantine approximation (section)

===General upper bound ===
{{main | Hurwitz's theorem (number theory)}}

The first important result about upper bounds for Diophantine approximations is [[Dirichlet's approximation theorem]], which implies that, for every irrational number {{math|''α''}}, there are infinitely many fractions <math>\tfrac{p}{q}\;</math> such that
: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{q^2}\,.</math>

This implies immediately that one cannot suppress the {{math|''ε''}} in the statement of Thue-Siegel-Roth theorem.

[[Adolf Hurwitz]] (1891)<ref>{{harvnb|Hurwitz|1891|p=279}}</ref> strengthened this result, proving that for every irrational number {{math|''α''}}, there are infinitely many fractions <math>\tfrac{p}{q}\;</math> such that
: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2}\,.</math>

Therefore, <math>\frac{1}{\sqrt{5}\, q^2}</math> is an upper bound for the Diophantine approximations of any irrational number.
The constant in this result may not be further improved without excluding some irrational numbers (see below).

[[Émile Borel]] (1903)<ref>{{harvnb|Perron|1913|loc=Chapter 2, Theorem 15}}</ref> showed that, in fact, given any irrational number {{math|''α''}}, and given three consecutive convergents of {{math|''α''}}, at least one must satisfy the inequality given in Hurwitz's Theorem.