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 $n$?
On
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 $r$.
COMMENT.-The inequality $\left|\pi -\dfrac qp\right|>\dfrac{1}{p^{42}}$, valid for all rational $\dfrac qp$ with $p≥2$), is not recent, it is due to Mahler (1903-1988) and was published in 1953 twice in The Philosophical Transactions of the Royal Society, 245 (England) and in Indagationes mathematicae, of the Royal Dutch Mathematical Society, 15 (Holland). This inequality, of an unusual character until the date of its discovery,("striking inequality" comments Alan Baker) indicates, with the limitations of its time, in what way rationals cannot approach $\pi$: every rational "close" to π determines its "distance from π", in the sense of never at a distance less than or equal to the power $42$ of its denominator.
Definition.- The measure of irrationality of a real α is the smallest of all reals $\mu$ for which there is a positive constant $A$ such that for all rational $\dfrac qp\ne \alpha$ with $p\gt 0$ we have $$|\alpha- \dfrac qp|\gt\dfrac{A}{p^{\space\mu}}$$ Theorem (Liouville).-For every algebraic number $\alpha$, real of degree $n≥2$ there exists a positive number $c(α)$ such that $\left|\alpha-\dfrac qp\right|>\dfrac{c(α)}{p^n}$ is true for all rational $\dfrac qp$ ($\space p>0$).
In other words, the measure of irrationality of the algebraic irrational $\alpha$ cannot be greater than its degree (it is quite less, but you had to wait until 1955 to find out). It was this theorem that allowed the Liouville transcendental numbers to be explicitly defined.
I give a succinct account of the progress in the value of $n$.
►Joseph Liouville (1809-1882), French: approximation $n+ϵ$, in 1844.
►Axel Thue (1863-1922), Norwegian: approximation $\dfrac n2 + 1 + ϵ$, in 1909.
►Carl Ludwig Siegel (1896-1981), German: approximation $2\sqrt n + ϵ$, in 1929. Siegel conjectured the best bound later found by Roth.
Freeman John Dyson (1923-), British-American: approximation $\sqrt{2n} + ϵ$, in 1947.
Klaus Friedrich Roth (1925-), German-British: $2 + ϵ$ approximation, in 1955. The best level, a profound result that earned Roth the Fields Medal. "The achievement is one that speacks for itself: it closes a chapter , and a new chapter is now opened. Roth’s theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated" (Harold Davenport, from his presentation by Roth for the Fields Medal at the Edinburgh International Congress, 1958).