As a curiosity and a test of the pseudo-random nature of trigonometric functions evalutated at integer-squares, I inquired to see what it would look like to graph the subset of $\mathbb{C}$ descibed by
$$\Big\{\sum_{k<t}e^{ik^2}\ |\ \forall t\in\mathbb{N}\Big\}$$
Or equiavently in $\mathbb{R^2}$
$$\Big\{\Big(\sum_{k<t}\cos k^2, \sum_{k<t}\sin k^2\Big)\ |\ \forall t\in\mathbb{N}\Big\}$$ Before diving into what I found graphically, which might be entertaining for some of you,
My main question is, has this set been studied before?
$\forall t < 500$, I was greeted by a euler-spiral/julia-set-resembling
In the above image, the absolute value of the minimum and maximum x and y coordinates of the entire graph is $20$ (with the center being the origin). Let's call this $scale=20$
$\forall t < 10^4, scale=20$, this spiral revolved into a noisy circle.
$\forall t < 10^5, scale=400$, this circle expanded into yet another euler-spiral-like shape, made up of smaller spiral segments.
$\forall t < 10^7, scale=2000$, an oddly symmetric jumble of more euler-spiral-like shapes spread out.
$\forall t < 10^8, scale=20000$, Lastly, this jumble proceeded to form into yet another euler-spiral-like shape.
This was as far as my computer and I had the patience for. It would seem to my eye that as $t\to\infty$, the set is quite fractal-like, creating larger and large euler-spiral-like shapes out of euler-spiral-like shapes of a lower order.
What do you make of this? What kind of properties of this set (e.g; it's fractal dimension) be formalized?