My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$.
My first question is why is it possible to state that up to a change of coordinates in $\mathbb{R}^{n}$ we may then suppose that the scalar product can be written as: $$\langle \xi- \eta,;x \rangle = (\xi_{1}-\eta_{1})x_{1}$$ where $x = (x_{1},...,x_{n})$ and $(\xi - \eta) = (\xi_{1}-\eta_{1},...,\xi_{n}-\eta_{n})$ and $\langle \cdot \rangle$ denotes the scalar product.
Secondly, if we consider the hypercube O = $H_{R}(x) = \prod_{i=1}^{n}(\alpha_{i,\beta_{i}}) = (\alpha_{1},\beta_{1}) \times \prod_{i=2}^{n}(\alpha_{i}, \beta_{i})$ and the following partition:
We divide $(\alpha_{1}, \beta_{1})$ into intervals of length $(\beta_{1}-\alpha_{1})2^{-m}$ where $m \in \mathbb{N}$. Each of these intervals is then subdivided into two parts of equal length $(\beta_{1}-\alpha_{1})2^{-(m+1)}$. The union of the first(resp. the second) subintervals is denoted by $I_{m}$(resp. $J_{m}$). We denote by $$O_{1}^{m} = I_{m} \times \prod_{i=2}^{n}(\alpha_{i}, \beta_{i})$$ $$O_{2}^{m} = J_{m} \times \prod_{i=2}^{n}(\alpha_{i}, \beta_{i})$$ then $\lim\limits_{m \rightarrow \infty} \text{meas } O_{1}^{m} = \lim\limits_{m \rightarrow \infty} \text{meas }O_{2}^{m} = \frac{1}{2}\text{meas } O$.
My question is how does it follow that if $x \in O_{1}^{m}$ then $\langle \xi- \eta;x \rangle \in I_{m}$?
Thanks for any assistance. Please ask if anything is unclear.
As noticed in the comments there are some inconsistency of notations in the question but I will try nevertheless to give an answer to your question. To do so I had to make some assumption on what is the precise context. I choose what seems to me to make sense. I hope it fits with what you really need.
For the first question: If the vectors $\xi$ and $\eta$ are fixed then $\xi-\eta$ is a fixed vector and we can build a base $(\xi-\eta,\zeta^{(2)},\zeta^{(3)},\ldots,\zeta^{(n)})$ of $\mathbb{R}^n$. In this base $\xi-\eta=(1,0,0,\ldots,0)$, so it is clear that $\langle \xi-\eta, x\rangle= (\xi_1-\eta_1) x_1$. Actually with this choice we even have $\langle \xi-\eta, x\rangle=x_1$.
Now for the second question I assume that the last equality holds. I guess that when you write $H_{R}(x) = \prod_{i=1}^{n}(\alpha_{i,\beta_{i}})$ you actually mean $H(\alpha,\beta) = \prod_{i=1}^{n}(\alpha_{i},\beta_{i}).$ I also guess that we consider the hypercube $$O=H(\eta,\xi)=\prod_{i=1}^{n}(\eta_{i},\xi_{i}).$$ Then $x\in O_{1}^{m} = I_{m} \times \prod_{i=2}^{n}(\eta_{i}, \xi_{i})$ implies $x_1\in I_m$. But by our assumptions this is precisely $\langle \xi-\eta, x\rangle\in I_m$.