TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ?
A little explanation about thisMore generaly the question would be : given an increasing sequence of integers $(u_k)$ and an irrational number $\alpha$, how do I tell if $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{2i\pi \alpha u_k}=0$ ? I'm not asking for a criterium for completely general sequences, an answer for sequences like $u_k=k^2$, $v_k=k!$ or $w_k=p(k)$ with $p\in \mathbf Z [X]$ would already be awesome.
A little explanation about this question :
In Real and Complex Analysis by Rudin there is the folowing exercise :
Let $f$ be a continuous, complex valued, $1$-periodic function and $\alpha$ an irrational number. Show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\alpha k)=\int_0^1f(x)\mathrm d x$. (We say that the $\left \lfloor{\alpha k}\right \rfloor $ are$(\alpha k)_k$ is uniformly distributed in $[0;1]$$\mathbf R / \mathbf Z$)
With the hint given by Rudin the proof is pretty straightforward : First one show that this is true for every $f_j=\exp(2i\pi j\cdot)$ with $j\in \mathbf{Z} $. Then using density of trigonometric polynomials in $(C^0_1(\mathbf{R}),\|\cdot\|_\infty)$ and the fact that the $0$-th Fourier coefficient of $f$ is it's integral over a period, one can conclude using a $3\varepsilon$ argument. This proof is possible because one can compute explicitly the sums $$\displaystyle \frac{1}{n}\sum_{k=0}^{n-1}e^{2i\pi j \alpha k}=\frac{1}{n}\cdot\frac{1-e^{2i\pi j\alpha n}}{1-e^{2i\pi j\alpha}}\longrightarrow 0 \text{ when }n\to\infty \text{ and }j\in \mathbf{Z}^*.$$
Now using a different approach (with dynamical systems and ergodic theorems) Tao show in his blog that the $\left \lfloor{\alpha k^2}\right \rfloor $ are$(\alpha k^2)_k $ is uniformly distributed in $[0;1]$$\mathbf R / \mathbf Z$ (corollary 2 in this blog). I'd like to prove this result using the methods of the exercice of Rudin, but this reduce to show that $$\displaystyle \frac{1}{n}\sum_{k=0}^{n-1}e^{2i\pi j \alpha k^2}\longrightarrow 0 \text{ when }n\to\infty \text{ and }j\in \mathbf{Z}^*.$$ Hence my question.
More generaly the question would be "given an increasing sequence of integersP.S. When i ask wolfram alpha to compute $(u_k)$$\sum_{k\geq0}e^{ik^2}$ it answer me with some particular value of the Jacobi-theta function. Of course the serie is there a criterium/method/necessarynot convergent but maybe it's some kind of resummation technique or sufficient condition/watheveranalytic continuation. I'm not familiar with these things but it might be interesting to tell if $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{2i\pi \alpha u_k}=0$ ?look in that direction.