Let G be a domain in $\mathbb{R}^3$ with the following properties:

  • bounded
  • finitely connected
  • positive Gaussian curvature
  • 2-times continuously differentiable boundary $\partial G$

Let $T>0$ and $G_T=\{Tx : x \in G \}$. We denote with $G^d_T$ the points with difference of less than $d>0$ from $G_T$ and write $G^d$ for the points with difference of less than $d>0$ from $G$.

I would like to compute the sum $$ \lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in (\mathbb{Z}^3\cap G_T^d\setminus G_T)}f(k), $$ where $f(k)$ is a function from $\mathbb{R}^3 $ to $\mathbb{R}$. In other words: I want to sum over every function value of $ f(k)$ for which $k\in \mathbb{Z}^3$ lies within the shell $G_T^d\setminus G_T$.

Some numerical experiments suggest that the number of points within the shell $G_T^d\setminus G_T$ depend on the thickness of the shell, the scaling factor T and the n-1 dimensional Hausdorff measure $\mu(\partial G)$ of $G$. More precisely $$ card \{k\in \mathbb{Z}^3: k\in G_T^d\setminus G^d \} \approx T^2 d \mu(\partial G) $$

It is clear that $$ \lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in (\mathbb{Z}^3\cap G_T^d\setminus G_T)} f(k)=\lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in \big(\frac{\mathbb{Z}^3}{T}\cap G^d\setminus G\big)} f(Tk). $$

My goal is to show that $$\lim_{T \rightarrow \infty}\frac{1}{T^2}\sum_{k\in \big(\frac{\mathbb{Z}^3}{T}\cap G^d\setminus G\big)} f(Tk) = \int_{G^d \setminus G}\lim_{T\rightarrow \infty}f(T\omega)d\omega . $$ Therefore i need to proof that the number of points inside the shell is indeed of order $T^2 d \mu(G)$ and more importantly that the points are equidistributed inside the region $G^d\setminus G^d$. My problem is that I'm not sure how to prove those two statements.

  • $\begingroup$ You should ask something more of $f$, because any function $g:\mathbb{Z}^3\to\mathbb{R}$ can be extended to an integrable function from $\mathbb{R}^3$ to $\mathbb{R}$, so the statement is false in general $\endgroup$ Jan 16 at 14:33
  • $\begingroup$ For $d<1$ this is essentially counting lattice points on the sphere itself, or in this case its weird squashed friend (maybe with a tiny variable center, probably $d$ small enough will make it centered at the origin). There are formulas for counting rational points on spheres (and probably ellipsoids as well). In particular you wouldn't assume they are normalized according to volume, but wrt surface measure... $\endgroup$
    – Asaf
    Jan 16 at 15:20
  • $\begingroup$ P.S. I don't know what $\mu(\partial G)$ means. Probably it means 0 in any reasonable case. You might want to consider $n-1$ dimensional Hausdorff measure or so... $\endgroup$
    – Asaf
    Jan 16 at 15:21
  • $\begingroup$ I think the way to actually calculate the sum you want is via Poisson summation (you might need to require a bit more smoothness over the boundary, unsure and probably want $f$ to be in some nice Sobolev space), together with some initial smoothing/regularization to make the related Bessel functions decay fast enough (just like in the circle problem say). $\endgroup$
    – Asaf
    Jan 16 at 15:28

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