Consider the circle $\mathbb{T}^1= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint subsegments $M\_j=[t\_j,t\_{j+1})$, $j = 0, \cdots, n$, $t\_0=t\_n$ and define the map $S$ by the formula
\begin{align}
S(t)=t+c\_j \quad \text{mod 1}, \quad t\in M\_j
\end{align}
Here $c\_j$ are real values. Such map is called interval translation map.


Consider sequences $\{t\_k^m\}$ and $\{c\_k^m\}$ of rational numbers that converge to $t\_k$ and $c\_k$ respectively. Let $S\_m$ be corresponding mappings.


Lemma: The approximating sequences $\{t\_k^m\}$ and $\{c\_k^m\}$ may be selected so that the following statement is true. Assume that numbers $i,j \in \{1,\cdots,n\}$: $i\neq j$, and $l\_1, \cdots ,l\_n \in \mathbb{Z}^+$ are such that
\begin{align}
t\_j-t\_i=\sum\limits\_{k=1}^nl\_kc\_k. \quad \*
\end{align}
Then for any $m \in \mathbb{N}$
\begin{align}
t\_j^m-t\_i^m = \sum\limits\_{k=1}^n l\_k c\_k^m
\end{align}


Proof: Let $L$ be the space of all values $t\_j$ and $c\_j$ satisfying all equations $\*$ that are true for the considered values of parameters. Since all coefficients $l\_k$ are integers, points with rational coordinates are dense in $L$.




---


My questions about the proof :


At first I emphasize that this lemma and its proof is in "Invariant measures for interval translations" written by S.kryzhevich.


I'm a little confused in this proof of the lemma.


If I understood right $L:=\{t\_j, c\_j ;\quad t\_j-t\_i=\sum\limits\_{k=1}^nl\_k c\_k\}$ which $t\_j \in M\_j$ and $c\_j$ could be any real numbers. Now since $l\_k$ are integers so $l\_k c\_k^m$ are still rational numbers even since $t\_j^m , t\_i^m$ are rational and since rational numbers are dense in $\mathbb{R}$ so points with rational coordinates are dense in $L$. Why being dense in $L$ guarantees the validation of the proof?



A lemma in approximating sequences
Let me answer your question.


The cxdimension of the space $L$ is finite (this is a subspace of ${\mathbb R}^N$ for some $N$.


Let us assume for simplicity that $codim L=1$ i.e. there is only one equation (\*). Then one of coordinates (let it be $c\_{n}$) can be expressed
as
$$c\_n=a\_1 t\_1+\ldots+ a\_n t\_n+b\_1 c\_1+\ldots b\_{n-1} c\_{n-1}$$
with all coefficients $a\_j$ and $b\_j$ being rational.


Then the converging sequence may be 'projected' to the space $L$ as follows: we replace the value $c\_n^{m}$ with
$$a\_1 t\_1^m+\ldots+ a\_n t\_n^m+b\_1 c\_1^m+\ldots b\_{n-1} c\_{n-1}^m.$$


The case $codim L >1$ can be considered similarly.


Sincerely yours,
Sergey


