I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't.
Very recently, I discovered the field "irregularity of distributions". Given a infinite discrete subset X of $\mathbb{R}^2$, the aim is to get a lower bound : $$|\operatorname{Card}(X \cap B) - |B|| \geq C f(B)$$
Here $f$ is a positive function, $C > 0$ is a constant, $|.|$ denote the 2-dimensional lebesgue measure and $B$ is a ball.
In this paper : http://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6400-11511_2006_Article_BF02392553.pdf, using Fourier techniques, it is proven (theorem 2A), that there exist a ball $B$ such that : $$|\operatorname{Card}(X \cap B) - |B|| \geq C |\partial B|^{1/2}$$
Here $|\cdot|$ on the right-hand side denotes the 1-dimensional lebesgue measure and $\partial B$ the boundary of $B$.
My problem is the following : Given $X$ an infinite discrete subset of $\mathbb{R}^2$, is it possible to find $\phi : \mathbb{R}^2 \to \mathbb{R}$, $C^{\infty}(\mathbb{R}^2)$ with compact support (or Schwartz), such that we get a non-trivial lower bound :
$$\left|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx \right| \geq C f(\phi)$$
Here $f$ is still a positive function.
The paper I cited is quite old, so do you have reference for this problem ?
Thanks,
Edit : I found this paper : https://arxiv.org/pdf/1307.2114.pdf, wich is very close to what I am looking for. The corollary 5.5 page 99 says that, for $\frac12 < r < \frac32$, there exists $c > 0$ such that for all $N \geq 2$ : $$\inf_{\{x_1, \dots, x_N\} \subset[0, 1(^2} \sup_{f \in W_0^{r, 2}[0, 1[^2} \Big|\frac1N \sum_{i = 1}^N f(x_i) - \int_{[0, 1(^2} f(x) dx \Big| \geq c \frac{\log^{1/2}(N)}{N^r}$$ where $W_0^{r, }[0, 1[^2$ consists in functions of the classical sobolev space $W^{r, 2}[0, 1[^2$, but such that all extension vanish if one coordinate is greater than 1.
This corollary is not sufficient for my application, in random point processes. I need something like (with condition on $r$) : $$\inf_{X} \sup_{K}\sup_{f \in W^{r, 2}(\mathbb{R}^2)} \Big|\sum_{x \in X \cap K} f(x) - \int_{K} f(x) dx \Big| \geq c F(\int_{\partial K} f(x) dx)$$ where $F$ is nome non negative function, the infimum in $X$ is taken over all the infinite discrete subsets in $\mathbb{R}^2$, and the infimum in $K$ is taken over translation, dilatation by a factor $\epsilon \in (0, 1]$, rotation on a fixed compact $K_0$.
Do you have any reference for this type of inequality ? Or, is it possible to derive this second type of inequality from the one above ?
