Dirichlet’s approximation

When you are first introduced to the real numbers you are almost forcefully told that they can be approximated by rationals numbers. The voice in your head goes something like ‘we can get as close as we like to an irrational by using rationals’. This is normally expressed in the statement \overline{\mathbb{Q}} = \mathbb{R} which reads \mathbb{Q} is dense in \mathbb{R} where ‘dense’ is a very precise mathematical statement.

It’s quite easy to think we are done here, but what invokes a good approximation? Dirichlet’s approximation is not only good in the sense you can control how close you want your rational approximation to be but it gives a splitting, in terms of behavior, with respect to rational and irrational numbers.

Theorem: For \alpha \in \mathbb{R} \backslash \mathbb{Q} we can find infinitely \frac{p}{q} \in \mathbb{Q} where \gcd(p,q) = 1 such that | \alpha - \frac{p}{q} | < \frac{1}{q^2}. Moreover, if \alpha \in \mathbb{Q} only finitely many approximations \frac{p}{q} exist.

Proof of theorem: We are fist going to introduce some notation. let x \in \mathbb{R} the integral part of x and the fractional part of x will be denoted and defined as [x] := \max \{z \in \mathbb{Z} : z \leq x \} and \{x\} := x - [x] respectively.

Let \alpha be irrational and Q be a positive integer.  The numbers 0, \{\alpha\}, \{2 \alpha \},..., \{Q\alpha \} gives us Q + 1 points between 0 and 1 that are distributed among the intervals [ \frac{j-1}{Q}, \frac{j}{Q} ) for j = 1,..., Q. Applying the pigeon hole principle means there has to be at least one interval containing at least two numbers \{k\alpha\} and \{l\alpha\} such that \{k\alpha\} \geq \{l\alpha\} with 0 \leq k, l \leq Q and k \neq l.

Computing the difference of \{k\alpha\} and \{l\alpha\} gives us \{k\alpha\} - \{l \alpha\} = k\alpha - [k\alpha] - l\alpha + [l\alpha] = \{(k-l)\alpha\} + [(k-l)\alpha] + [l\alpha] - [k\alpha]  which lies in the interval [0, \frac{1}{Q} ) meaning that [(k-l)\alpha] + [l\alpha] - [k\alpha] = 0 as the LHS is an integer. Setting q = k - l and p = [q \alpha] gives the property \{q \alpha\} < \frac{1}{Q} and so computing the difference between \alpha and \frac{p}{q} gives | \alpha - \frac{p}{q} | = \frac{|q\alpha - p |}{q} = \frac{\{q\alpha\}}{q} < \frac{1}{Qq}< \frac{1}{q^2} . The last inequality follows from q < Q .

To see the split in behaviour we first assume \alpha is irrational and that there are finitely many approximations \frac{p_1}{q_1},..., \frac{p_n}{q_n} . Since \alpha is irrational by the fact $latex  \overline{\mathbb{Q}} = \mathbb{R}$ we can fine Q (a positive integer) such that for all j = 1,...,n we have |\alpha - \frac{p_j}{q_j} | > \frac{1}{Q} which contradicts the established facts above.

In the second case when \alpha is a rational number, say \alpha = \frac{a}{b} with a \in \mathbb{Z} and b\in \mathbb{N} if \frac{a}{b} \neq \frac{p}{q} then by direct computation |\alpha - \frac{p}{q} | = \frac{|aq - bp|}{bq} \geq \frac{1}{bq} so we can only have finitely many such approximations as we require q < b .

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