Say I have a real number $x>0$ and some constant $C>0$. I want to use a rational number $q$ such that the following holds:
$|x-q|<Cx$
Of course such numbers $q$ exist. What I want is to find a bound on the height $H$ needed to find such a rational, in terms of $x,C$. (Height of a rational number is the maximum between the absolute values of its nominator and denominator in reduced form).
Does anyone have a reference for such approximation problems or an idea how to solve this one? I am trying to make this inequality similiar to a dirichlet type inequality but can't manage to do this.
Thank you!