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Changing short description from "Rational-number approximation of real numbers" to "Rational-number approximation of a real number" (Shortdesc helper) |
Hurwitz proved infinitely many p/q satisfy |a - p/q| < 1/(sqrt(5)*q^2). Borel proved that given three successive convergents, at least one must satisfy this inequality. |
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This implies immediately that one cannot suppress the {{math|''ε''}} in the statement of Thue-Siegel-Roth theorem.
: <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.
=== Equivalent real numbers ===
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