I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ with $0<q\le Q$ such that $|x-p/q|<1/qQ$.


I am looking for a version of this theorem where $q$ is only a prime power, say $2^n$. Then by binary approximation of real number one can immediately say that $|x-p/2^n|<1/2^n$. But can we do better along the line of Dirichlet?


Thanks for any reference.



Dirichlet's approximation only using prime power as denominator
You cannot do much better than what you observed, because the fractional parts $\{2^n x\}$ are essentially the tails in the binary expansion of $x$, so they can be bounded away from zero (even for rational numbers $x$).


However, Furstenberg (1967) proved that the fractional parts $\{2^m3^n x\}$ are dense in $(0,1)$ for any irrational $x$, and in fact the same holds for any nonlacunary semigroup. A simple proof was given by Boshernitzan (Proceedings of the AMS, 1994).


