Hurwitz's Theorem (Number Theory)

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Theorem

Let $\xi$ be an irrational number.

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

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


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:

$\left|{\xi - \dfrac p q}\right| < \dfrac 1 {A \, q^2}$

$\Box$


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 there are an infinite number of approximations that satisfy the given inequality.

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

$\blacksquare$


Source of Name

This entry was named for Adolf Hurwitz.


Sources