given an irrational $x > 0$, approximate it by a rational : $$x = \frac{p}{q} + \epsilon$$
the residual can be seen as a function of $q \in \mathbb{N}$ : $$\epsilon(q) = x - \frac{\lfloor q x\rfloor}{q}$$
what will be the order of the function $q \to q\,\epsilon(q)$ ? we know that $$\lim\inf_{q \to \infty} q \,\epsilon(q) = 0$$
because $\{a+bx \ |\ a,b \in \mathbb{Z}\}$ is dense in $\mathbb{R}$, but can we get things like $$\lim\inf_{q \to \infty} \,\epsilon(q) q^k = 0$$ for some $k > 1$. apparently, as pointed out by Gerry, Dirichlet's approximation theorem says that $k = 2$ can be proven, while the Thue–Siegel–Roth_theorem says that $2$ is the maximum for algebraic irrationals...
And I was also wondering if this could be related to the Farey sequence ?
If $\liminf f(q)=0$, then surely $\liminf f(q)/\log q=0$ as well, so maybe that's not what you meant to ask. Anyway, Dirichlet's Theorem on Diophantine Approximation says that if $x$ is irrational then there are infinitely many pairs of integers $p,q$ such that $|qx-p|<q^{-1}$, so that should give you what you want (once you figure out what you want).