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Let $S:=\{ x\in \mathbb{R}^d:||x||_2=1\}$ be the d-dimensional unit sphere, where $||x||_2$ is the euclidean norm.

Let $\epsilon>0$ and $s\in S$ be an arbitrary point on the sphere.

Is it correct that there exists an $\alpha>0$ and a $k\in \mathbb{Z}^d\setminus\{0\}$ such that the distance between $\alpha s-k$ is less than $\epsilon$?

In other words can i scale every point on the unit sphere such that the distance to an non-zero integer is arbitrarily small

HyyFly
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    This genre of question is very interesting, and possibly subtler than some people might have supposed. If no one who has a sharper answer responds after a while, I will tell what I know.... such as it is. :) – paul garrett Oct 22 '21 at 22:08
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    I think this is related to Kronecker's theorem – tzndls Oct 22 '21 at 22:53
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    A quick comment, along the lines of @GabrielC.Barbosa's comment: for situations more complicated than products of circles, Kronecker's theorem becomes "Weyl's criterion" for equidistribution. Kloosterman and others in the early 20th century treated this issue on spheres, in various dimensions, with odd ambient dimensions being more complicated, as half-integer-weight modular forms enter, etc. More later... – paul garrett Oct 22 '21 at 23:34

1 Answers1

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The statement is true for any vector $v\in\mathbb R^d$, not necessarily on the unit sphere. For any $x\in\mathbb R^d$, let $x\text{ mod }\mathbb Z^d$ denote the unique element in $y\in [0, 1)^d$ such that $x-y\in\mathbb Z^d$.

Because $[0, 1)^d$ is bounded, the sequence $\{nv \text{ mod } \mathbb Z^d\mid n\in\mathbb N\}$ has a convergent subsequence $\{n_iv\text{ mod } \mathbb Z^d \mid i\in\mathbb N\}$. In particular, for large enough $i>j$, there is $\|n_i v \text{ mod } \mathbb Z^d - n_j v \text{ mod } \mathbb Z^d \|= \|(n_i-n_j) v \text{ mod } \mathbb Z^d\|<\epsilon$. That is $(n_i-n_j)v$ is very close to an integral point, and we may take $\alpha = n_i - n_j$ (in particular $\alpha$ can be chosen as an integer).

Note that $\alpha$ can be taken to be as large as needed, and when $\alpha$ is large, $\|\alpha v\|$ is not close to $0$, so the $k$ can always be taken to be nonzero.

Also the same argument applies to any lattice with compact quotient, not necessarily $\mathbb Z^d$.

Just a user
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