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For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\cdot R^d$, as well as the number of distinct 1-dimensional linear spaces they generate (with a different constant, of course, both constant are well-known).

Is it correct that they generate (at least) $c(k, d)\cdot R^{kd}$ distinct $k$-dimensional planes for all $k=1,2,\ldots,d-1$ with some positive constants $c(k, d)$?

A connected question: is it correct that the hyperplanes orthogonal to these vectors partition the space onto at least $C(d)\cdot R^{d(d-1)}$ open regions?

If yes (what I expect to be the case and possibly even have a proof, but quite tedious and with unpleasant details), are the sharp constants computed and what are the simplest proofs?

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For $k=d-1$ this is a result of Bárány-Harcos-Pach-Tardos (2001). See Theorem 3 in the preprint version or the published version.

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    $\begingroup$ The case $k=d-1$ implies the general case $0 < k < d$ since the number of $d-1$-dimensional hyperplanes is at most the number of $k$-dimensional hyperplanes times the number of points in the ball to the power of $d-1-k$. $\endgroup$
    – Aleksei Kulikov
    Commented Jul 15 at 7:54
  • $\begingroup$ @AlekseiKulikov Excellent point! $\endgroup$
    – GH from MO
    Commented Jul 15 at 15:12

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