TITLE: Can we get good rational approximations in all residue classes?
QUESTION [16 upvotes]: The classic Hurwitz theorem for rational approximations (in simplest form; the constant can of course be improved) gives infinitely many approximations $\frac mn$ to an irrational $\alpha$ with $|\frac mn-\alpha|\lt\frac1{n^2}$.  Just recently, in trying to answer a question related to rational approximation of $\pi$ I tripped over a limitation of this theorem: it tells us nothing about the specific $m,n$ of an approximation.  I'm interested in $n$ particularly, and wondering if there are any 'Dirichlet-style' results that say that for any irrational $\alpha$ and for any $a, d$ we can get good approximations (in the sense above) with $n\equiv a\pmod d$.  Is this a known result?

REPLY [11 votes]: Just to make sure that it's not hidden in the comments for future readers, I want to point out that the paper "On the Approximation of Irrational Numbers With Rationals Restricted By Congruence Relations" actually gives a positive answer to a slight extension of my question (where rather than just having $\left|\frac mn-\alpha\right|\lt \frac1{n^2}$ one has that the difference is bounded for $\frac{C}{n^2}$ for some $C$ possibly dependent on the congruence parameters), which was actually sufficient to show what I was originally after.  The precise statement is:

For any irrational $\xi$, any $s\geq 1$, and integers $a, b$, there
  are infinitely many integers $m, n$ satisfying
  $\displaystyle\left|\xi-\frac{m}{n}\right|\lt\frac{2s^2}{n^2}$,
  $m\equiv a\pmod s$, $n\equiv b\pmod s$.