Bound on the approximation of a number on the unit interval by the positive multiples of an irrational number

Question

Suppose that I am given an irrational $\alpha$, then the set $$\{\{n\alpha\}:n\in \mathbb N\}$$

with $\{x\}=x-\lfloor x\rfloor$ is dense in $[0,1]$ (see [1]).

Now given $\epsilon>0$ and some target number $0<p<1$, I would like to find an upper bound on $n\in\mathbb N$ that verifies $$\left|p-\{n\alpha\}\right|<\epsilon$$

Any ideas appreciated. There are many related questions. For instance:

[1] Density of positive multiples of an irrational number

[2] Multiples of a given irrational number can be arbitrarily close to a natural number

[3] Dense set in the unit circle- reference needed