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.

  • $\begingroup$ @Sorry I made a mistake . It was meant to read $|\alpha-\tfrac{p}{q}|>\tfrac{1}{Cq^2}.$ Is there a theorem which is relevant to this inequality similar to Hurwitz ? $\endgroup$ – bhapi Apr 24 at 4:37
  • $\begingroup$ Did you see my comment on your other question, math.stackexchange.com/questions/3191224/… $\endgroup$ – Gerry Myerson Apr 24 at 5:30
  • $\begingroup$ Ah no I hadn't actually , thank you. I followed it up until P' arises when trying to evaluate $(p/q-\alpha)$ which we uses to get C. I dont see how we can use P' to estimate it or what relevance estimating it has to finding C. The advice you gave above was great it really cleared up pretty much all of my confusion. Its just those two small points I just cant get... $\endgroup$ – bhapi Apr 24 at 5:51
  • $\begingroup$ I have posted an answer to the other question. Please see whether it helps. $\endgroup$ – Gerry Myerson Apr 26 at 4:42
  • $\begingroup$ Are you still here? $\endgroup$ – Gerry Myerson Apr 27 at 7:45

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