I've seen many examples of Dirichlet's approximation being proven , or other questions regarding to the theory of the approximation on this site and others but I would like to see a concrete example of it actually being used.
For example if we were given some quadratic irrational $\alpha = \frac{11+\sqrt{2605}}{18}$ , I would like to see examples of how to find rationals $\tfrac{p}{q}$ which satisfy $|\alpha-\tfrac{p}{q}|<\tfrac{1}{q^2}$.
Or similarly given $|\alpha-\tfrac{p}{q}|<\tfrac{1}{Cq^2}$, what constant C satisfies this equation for all rationals, and other such concrete applications as this .
Could anyone present any links to where I might be able to learn more about how to tackle such problems ? Many thanks in advance.
The usual proof of Dirichlet is constructive. It shows you how to find $p,q$. Given $\alpha$, pick a positive integer $N$, and calculate the $N+1$ numbers $\{0\alpha\},\{1\alpha\},\{2\alpha\},\dots,\{N\alpha\}$, where $\{x\}$ denotes the fractional part of $x$. You are now looking at $N+1$ numbers in the interval $[0,1)$, so two of them must differ by less than $1/N$; say, $\{r\alpha\}-\{s\alpha\}<N^{-1}$. This gives $\{q\alpha\}<N^{-1}$ where $q=|r-s|$, $|q\alpha-p|<N^{-1}$ where $p$ is the integer part of $q\alpha$, $|\alpha-(p/q)|<(qN)^{-1}\le q^{-2}$.
But a more efficient way to find such $p,q$ goes by way of the continued fraction expansion of $\alpha$.
Finding $C$ which satisfies $|\alpha-(p/q)|<(Cq^2)^{-1}$ for all rationals, doesn't make any sense. What is true is that for any real irrational $\alpha$ there are infinitely many rationals $p/q$ such that $|\alpha-(p/q)|<(\sqrt5q^2)^{-1}$. That's Hurwitz' Theorem, and again the best approach is via continued fractions rather than through Dirichlet. Any good Number Theory text should set you straight on this.