Hurwitz's Theorem (Number Theory)

This proof is about Hurwitz's Theorem in the context of Number Theory. For other uses, see Hurwitz's Theorem.

Theorem

Let $\xi$ be an irrational number.

Then there are infinitely many relatively prime integers $p, q \in \Z$ such that:

$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$

Proof

Lemma 1

Let $\xi$ be an irrational number.

Let $A \in \R$ be a real number strictly greater than $\sqrt 5$.

Then there may exist at most a finite number of relatively prime integers $p, q \in \Z$ such that:

$\size {\xi - \dfrac p q} < \dfrac 1 {A \, q^2}$

$\Box$

Lemma 2

Let $\xi$ be an irrational number.

Let there be $3$ consecutive convergents of the continued fraction to $\xi$.

Then at least one of them, $\dfrac p q$ say, satisfies:

$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$

$\Box$

There are an infinite number of convergents to $\xi$.

Taking these in sets of $3$ at a time, it can be seen from Lemma 2 that at least one of them satisfies the given inequality.

From Lemma 1 it is seen that this inequality is the best possible.

$\blacksquare$

Also see

• Liouville's Theorem (Number Theory)

Source of Name

This entry was named for Adolf Hurwitz.

Sources

• 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.): $11.8$: The measure of the closest approximation to an arbitrary irrational

• Dec. 2002: Manuel Benito and J. Javier Escribano: An Easy Proof of Hurwitz's Theorem (Amer. Math. Monthly Vol. 109, no. 10: pp. 916 – 918)  www.jstor.org/stable/3072459