Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some big $M>0$ what type of bound can one get on $$\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \cdot \langle v, w \rangle),$$ where $B(0,M)$ is the euclidean ball of radius $M$ and center $0$? I believe it is something like $\big(\varepsilon\frac{M}{\sqrt{d}}\big)^d$ for $\varepsilon\to 0$ when $M$ grows, since this is the type of behaviour when we sum over the cube instead of the euclidean ball. Is there some general equidistribution theorem one might be able to apply here?
1

$\begingroup$ What are $\epsilon$ and $n$? $\endgroup$ – Anthony Quas May 1 at 4:55

$\begingroup$ Oh sorry $n$ was meant to be $d$, $\varepsilon$ should be some function of $M$ going to $0$ as $M$ goes to infinity. $\endgroup$ – Marcelo Campos May 1 at 5:03

$\begingroup$ So for cubes, this does not even converge to zero... $\endgroup$ – Anthony Quas May 1 at 5:22

$\begingroup$ Hmm isn't it true that for cubes $[M,M]^d$ we have $$\frac{1}{M^d}\sum_{w\in \mathbb{Z}^d\cap [M,M]^d}\exp\big(2\pi i\eta \cdot \langle v, w \rangle\big)\to 0,$$ when $M$ goes to infinity, just by using equidistribution of irrational rotations modulo 1? $\endgroup$ – Marcelo Campos May 1 at 14:47

$\begingroup$ There was no averaging in the question. But in that case, it scales like $C/M^d$. $\endgroup$ – Anthony Quas May 1 at 15:18