Corollary 21.7.4`CF4226A5034F`

Corollary 21.7.4.CF4226A5034F Let $K\subset M_{\mathbb R}$ be a convex body, and suppose that subsets $S_ k\subset M$ satisfy

\[ kK\cap M\subset S_ k\subset (kK)^{r_ k}\cap M, \qquad r_ k\geq 0,\quad r_ k/k\to 0. \]

Then $k^{-n}\# S_ k\to \operatorname {vol}(K)$. In particular, a boundary error of bounded size does not affect the leading term of the count, even when $K$ has zero volume.

Proof. By theorem 21.7.3 with $f=1$ and $a_ k=0$, both $k^{-n}\# (kK\cap M)$ and $k^{-n}\# ((kK)^{r_ k}\cap M)$ tend to $\operatorname {vol}(K)$, and $\# S_ k$ lies between these two counts. $\square$

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BibTeX citation

@misc{pome-tag-CF4226A5034F,
  author = {{Pome contributors}},
  title = {Corollary 21.7.4},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/CF4226A5034F/}
}

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Corollary 21.7.4CF4226A5034F

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Corollary 21.7.4`CF4226A5034F`