We set $A_{k} = \frac{X}{\sqrt{k}}$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $\varepsilon$ be in $]0, 1[$ and $\delta$ be in $]0, 1/2[$. We assume that $k \geqslant 160 \frac{\ln(1/\delta)}{\varepsilon^{2}}$. Let $v_{1}, \ldots, v_{N}$ be distinct vectors in $\mathbb{R}^{d}$. For every $(i, j) \in \llbracket 1, N \rrbracket^{2}$ such that $i < j$ we denote by $E_{ij}$ the event $$(1 - \varepsilon) \|v_{i} - v_{j}\| \leqslant \|A_{k} \cdot v_{i} - A_{k} \cdot v_{j}\| \leqslant (1 + \varepsilon) \|v_{i} - v_{j}\|$$ Show that $\mathbb{P}\left(\overline{E_{ij}}\right) < \delta$, where $\overline{E_{ij}}$ denotes the complementary event of $E_{ij}$.
We set $A_{k} = \frac{X}{\sqrt{k}}$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $\varepsilon$ be in $]0, 1[$ and $\delta$ be in $]0, 1/2[$. We assume that $k \geqslant 160 \frac{\ln(1/\delta)}{\varepsilon^{2}}$. Let $v_{1}, \ldots, v_{N}$ be distinct vectors in $\mathbb{R}^{d}$. For every $(i, j) \in \llbracket 1, N \rrbracket^{2}$ such that $i < j$ we denote by $E_{ij}$ the event
$$(1 - \varepsilon) \|v_{i} - v_{j}\| \leqslant \|A_{k} \cdot v_{i} - A_{k} \cdot v_{j}\| \leqslant (1 + \varepsilon) \|v_{i} - v_{j}\|$$
Show that $\mathbb{P}\left(\overline{E_{ij}}\right) < \delta$, where $\overline{E_{ij}}$ denotes the complementary event of $E_{ij}$.