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?



Can we get good rational approximations in all residue classes?
The answer is no. Take $\alpha=\sqrt{2}$ and note that if $|\sqrt{2}-m/n|\le 1/n^2$ then we have $0<|2n^2-m^2| \le (\sqrt{2}n+m)/n \le 3$. Now suppose we want $n\equiv 4\pmod p$ say. Then we must have that $32-m^2 \equiv b \pmod p$ for some $|b|\le 3$. But we can find a prime $p$ for which the numbers $29$ to $35$ are all quadratic non-residues $\pmod p$ (for example, choose $p$ so that $2, 7, 11, 29, 31$ are all non-residues $\pmod p$, and $3, 5, 17$ are residues). Thus there are no good approximations to $\sqrt{2}$ with $n\equiv 4\pmod p$ for such a prime $p$. One can clearly vary this argument a fair bit. 


