Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group. 



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> **Problem 1.** Is it true that for any finite set $F\subset\mathbb T^\omega$ and any neighborhood $U\subseteq \mathbb T^\omega$ of zero there exists an automorphism $\alpha$ of $\mathbb T^\omega$ such that $\alpha(F)\subset U$?
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This problem can be reformulated in the language of the special linear groups $SL(n,\mathbb Z)$.



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> **Problem 2.** Is it true that for any $n\in\mathbb N$, neighborhood of zero $U$ in $\mathbb R^n$ and vectors $x\_1,\dots,x\_n$ in $\mathbb R^\omega$ there exists $m>n$ and a matrix $A\in SL(m,\mathbb Z)$ such that $\mathrm{pr}\_n\circ A\circ \mathrm{pr}\_m(x\_i)\in U$ for all $i\in\{1,\dots,n\}$?
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Here $\mathrm{pr\_k}:\mathbb R^\omega\to\mathbb R^k$, $\mathrm{pr}\_k:x\mapsto x{\restriction}k$, is the projection onto the first $k$ coordinates.


**Remark 1.** For any field $\mathbb F$ and vectors $x\_1,\dots,x\_n\in\mathbb F^{2n}$ there exists a linear transformation $A\in SL(2n,\mathbb F)$ of $\mathbb F^{2n}$ such that $A(\{x\_1,\dots,x\_n\})\subset\{0\}^n\times\mathbb F^n$.


**Remark 2.** For any vector $(x,y)\in\mathbb R^2$ and any $\varepsilon>0$ there exists a matrix $A\in SL(2,\mathbb Z)$ such that $(x,y)\cdot A\in (-\varepsilon,\varepsilon)\times\mathbb R$. Such matrix $A$ can be constructed by finding relatively prime integer numbers $p,q$ such that $|xp+yq|<\varepsilon$ and then finding integer numbers $a,b$ such that $pb-qa=1$ (using the extended Euclidean algorithm). Then the matrix $A=\left(\begin{array}&p&a\\q&b\end{array}\right)\in SL(2,\mathbb Z)$ has the required property.


Taking into account Remarks 1 and 2, I would expect that the following stronger form of Problem 2 has an affirmative answer.



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> **Problem 3.** Is it true that for any $n\in\mathbb N$ and vectors $x\_1,\dots,x\_n\in\mathbb R^{2n}$ there exists a linear transformation $A\in SL(2n,\mathbb Z)$ such that $A(\{x\_1,\dots,x\_n\})\subset(-\varepsilon,\varepsilon)^n\times\mathbb R^n$?
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How flexible is the infinite-dimensional torus?
This is a draft proof of an affirmative answer to Problem 3. 


**Proposition.** For any $n\in\mathbb N$, $\varepsilon>0$ and vectors $x\_1,\dots,x\_n\in\mathbb R^{2n}$ there exists a linear transformation $A\in SL(2n,\mathbb Z)$ such that $A(\{x\_1,\dots,x\_n\})\subset[-\varepsilon,\varepsilon]^n\times\mathbb R^n$.


*Proof*. Let $m=2n$, $x\_i=(x\_{i1},\dots,x\_{im})$ for each $i$ and $X=\|x\_{ij}\|$. We shall call a column
of a matrix *small*, provided all its entries have absolute value at most than $\varepsilon$, and *big*, otherwise. Let $k$ be the maximal number of small columns in a matrix $XB$, where $B\in SL(m,\mathbb Z)$ and $C$ be an arbitrary matrix in $SL(m,\mathbb Z)$ such that $XC$ has $k$ small columns. It suffices to show that if $k<n$ then there exists a matrix $D\in SL(m,\mathbb Z)$ such that a matrix $XCD$ has $k+1$ small columns. Without loss of generality we can suppose that the big columns $y\_1,\dots, y\_{l}$ of the matrix $XC$ are from the first to the $l$-th. Pick a positive integer $M$ such that the maximal absolute value of an entry of these columns is at most $M\varepsilon$. Pick an arbitrary positive integer $K>(2lM)^n$ and define a map $f$ from the subset $Q^l$ of points of the set $[-K, K]^l$ with all integer coordinates to $\mathbb R^n$ by putting $f(d)=d\_1y\_1+\dots + d\_ly\_l$ for each $d=(d\_1,\dots,d\_l)\in Q^l$. Since all $|d\_i|\le K$ and all entries of columns $y\_i$ have absolute value at most $M\varepsilon$, each coordinate of a vector $f(d)$ is at most $lKM\varepsilon$. Therefore the image $f(Q^l)$ can be covered by $(2lKM)^n$ axis-parallel cubes with side $\varepsilon$. Since $|Q^l|=(2K+1)^l>(2K)^{n+1}>(2lKM)^n$, there exist two distinct elements $d’$ and $d’$ of $Q^l$ such that each coordinate of a vector $y=f(d’)-f(d’’)$ has an absolute value at most $\varepsilon$. Put $d=d’-d’’$. Dividing entries of $d$ by their greatest common divisor, if needed, we can assume that the greatest common divisor of entries of $d$ is $1$. It is well-known (see, for instance, [this](https://math.stackexchange.com/questions/2292909/integers-with-greatest-common-divisor-1-fit-in-a-row-of-invertible-matrix) MSE thread]) that there exists a matrix $D’\in SL(l,\mathbb Z)$ whose first column is $d$. Put $D=\begin{pmatrix} D’ & 0\\ 0 & I\end{pmatrix}$, where $I$ is the $(m-l)\times (m-l)$ identity matrix. Then the first column of the matrix $XCD$ is small, whereas its last $k$ columns are the same as in the matrix $XC$. $\square$


