The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value.
However, this problem becomes particularly annoying to deal with when trying to estimate the discrete length functional of a function $f(x)$, given by,
$$(1) \quad L^b_a[f(x)]=\frac{b-a}{\omega} \cdot \sum_{n=1}^{\omega} \sqrt{1+\omega^2 \cdot \left(f \left(\frac{(b-a)\cdot n}{\omega} \right)-f\left(\frac{(b-a)\cdot (n-1)}{\omega}\right) \right)^2}$$
Given,
$$(2) \quad f(x)=\sum_{k=1}^{\infty} k^{-p} \cdot \sin(k^p \cdot x)$$
I want to know the asymptotic behavior of $(1)$ as $\omega \rightarrow \infty$. Specifically, I want the techniques used to be clearly explained and possibly reproducible for other functions.
Because of Gibb's Phenomenon, I can't fully trust my numerical results that find $L \sim \omega^{2^{-p}}, \ p \gt v$
However, I do have formal methods pointing towards a power law with the exponent $\beta$ being bounded by,
$$(3) \quad 0\lt \beta \lt 1/p$$
Motivation
The motivation behind the problem is to find the box dimension of the graph of $f(x)$. So a good criteria for a good answer is whether or not it is accuarate enough to yield this number. As mentioned in comments, I don't currently know of a way to approximate the sum except in special cases. However,
$$\sqrt{1+x^2} \sim |x|$$
Is useful since, the derivatives involved are much larger than unity.
(I intend to make this a bounty and add more information)