Created by potrace 1.14, written by Peter Selinger 2001-2017 Eύρηκω

Liouville’s theorem on diophantine approximation

 
  • Sep 09, 2021

Liouville’s theorem on diophantine approximation Let \( \alpha \) be an irrational number that is algebraic of degree \( d \). Then for any real number \( e > d \), there are at most finitely many rational numbers \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \).

If there is no such rational number \( \frac { { p } } { q } \) then the number of solution is clearly finite.

Now assume there exists at least one rational number \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \).

Let \( P(x) \) be a polynomial of degree \( d \), with integers coefficients, such that \( P(\alpha)=0 \). Choose \( \epsilon \) such that \( P(x) \) has no roots other that \( \alpha \) on the interval \( [\alpha - \epsilon, \alpha + \epsilon] \)

Write \( P(x)=(x-\alpha) \cdot Q(x) \) where \( Q(x) \) is a monic polynomial with real coefficients of degree \( d-1 \). Since \( Q(x) \) is continuous, there exists \( K>0 \) such that \( |Q(x)| \leq K \) on the interval \( [\alpha - \epsilon, \alpha + \epsilon] \)

For all rational number \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \), \( \left| \frac { p } { q } - \alpha \right|>\epsilon \) or \( \left| \frac { p } { q } - \alpha \right| \leq \epsilon \)

For all rational number \( \frac { { p } } { q } \) such that \( \epsilon < \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \). \( q^e< 1/\epsilon \) and \( p \in [q(\alpha - \frac { 1 } { q ^ { e } }), q(\alpha + \frac { 1 } { q ^ { e } })] \), it is not difficult to tell the number of such rational numbers will be finite.

For all rational number \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \, \land \, \left| \frac { p } { q } - \alpha \right| \leq \epsilon \). Note

\[ \left| P \left( \frac { p } { q } \right) \right| = \left| \frac { p } { q } - \alpha \right| \left| Q \left( \frac { p } { q } \right) \right| < \frac { 1 } { q ^ { e } } K \]

Since \( P \) has degree \( d \) and integer coefficients,

\[ P ( \frac { p } { q } )=m_d\frac { p^d } { q^d }+m_{d-1}\frac { p^{d-1} } { q^{d-1} }+\cdots+m_0=\frac{m_d p^d+ m_{d-1} p^{d-1} q+\cdots+m_0q^d}{q^d}, \]

where both the nominator and the denominator are integers. Since \( \alpha \) is irrational, \( P ( \frac { p } { q } ) \neq 0 \). Then \( \left| P \left( \frac { p } { q } \right) \right| \geq \frac { 1 } { q ^ { d } } \). Note

\[ \frac { 1 } { q ^ { d } } \leq \left| P \left( \frac { p } { q } \right) \right|< \frac { 1 } { q ^ { e } } K \implies q^{e-d}<K \]

Similarly, the number of rational numbers \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \, \land \, \left| \frac { p } { q } - \alpha \right| \leq \epsilon \) will be finite.

Therefore, there are at most finitely many rational numbers \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \).

Search