# Number of lattice points in homotetic image

I asked this question on MSE a week ago and it gave me a tumbleweed badge :-)

Let $\Lambda$ be a lattice in $\mathbb R^n$, with covolume $\Gamma$. Moreover, let $S$ be a bounded (Lebesgue-)measurable subset of $\mathbb R^n$ and for each $\alpha > 0$ define $\alpha S := \{\alpha \mathbf x : \mathbf x \in S\}$.

Under which (reasonable) hypothesis on $S$ is it true that $$\lim_{\alpha \to +\infty} \frac{\#(\alpha S \cap \Lambda)}{\alpha^n \Gamma} = \mu(S) \; ?$$ Here $\mu$ is the Lebesgue measure on $\mathbb R^n$.

Thank you for any suggestion/reference.

NOTE 1: As pointed out by Igor Rivin, at the top of the 9th page of

I. Kapovich, I. Rivin, P. Schupp, and V. Shpilrain, Densities in free groups and $\mathbb{Z}^k$, Visible Points and Test Elements, Mathematical Research Letters 14 (2007), pp. 263--284.

it is written that for $\Lambda = \mathbb{Z}^n$ and $S$ bounded, open, and with smooth piecewise boundary the claim is well-known to be true.

[2]:

• In the case that $S$ is a polytope, much more precise statements are known. Search for "Ehrhart theory". – Emanuele Tron Dec 22 '15 at 21:16