Does the function have a positive lower bound? Let $\zeta$ be an irrational number and $p, q, n$ be natural numbers. The function $g$ is defined as:
$$g_n(\zeta,p,q)=p^n|p\zeta-q|.$$
Does there exist a positive $r$ such that:
$$\forall p\in\mathbb{N}-\{0\},\ \forall q\in\mathbb{N},\ g_n(\zeta,p,q)\geq r?$$
Is the result relevant to $\zeta$ and $n$?

There is no lower bound.
Consider an increasing sequence $e_i$ of positive integers and let $b$ be a positive integer. Then
$$\zeta = \sum_i b^{-e_i}$$
will be irrational if the $e_i$ are chosen not to result in a periodic tail for the base-$b$ expansion of $\zeta$.
But $$\min_q g_n(\zeta, b^{e_k}, q) = \sum_{i > k}b^{(n+1)e_k-e_i}$$
and no matter what $n, k, r$ we choose, there will be sequences with $e_{k+1}$ large enough to make that remainder smaller than $r$.
