We keep the notations and hypotheses from above. Let $v_{1}, \ldots, v_{N}$ be distinct vectors in $\mathbb{R}^{d}$. 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 with independent Rademacher coefficients. Let $\varepsilon \in ]0,1[$, $\delta \in ]0, 1/2[$, and $k \geqslant 160\frac{\ln(1/\delta)}{\varepsilon^{2}}$. For every $(i, j) \in \llbracket 1, N \rrbracket^{2}$ such that $i < j$, $E_{ij}$ denotes the event $$(1 - \varepsilon)\|v_{i} - v_{j}\| \leqslant \|A_{k} \cdot v_{i} - A_{k} \cdot v_{j}\| \leqslant (1 + \varepsilon)\|v_{i} - v_{j}\|$$ Deduce that $\mathbb{P}\left(\bigcap_{1 \leqslant i < j \leqslant N} E_{ij}\right) \geqslant 1 - \frac{N(N-1)}{2}\delta$.
We keep the notations and hypotheses from above. Let $v_{1}, \ldots, v_{N}$ be distinct vectors in $\mathbb{R}^{d}$. 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 with independent Rademacher coefficients. Let $\varepsilon \in ]0,1[$, $\delta \in ]0, 1/2[$, and $k \geqslant 160\frac{\ln(1/\delta)}{\varepsilon^{2}}$. For every $(i, j) \in \llbracket 1, N \rrbracket^{2}$ such that $i < j$, $E_{ij}$ denotes the event
$$(1 - \varepsilon)\|v_{i} - v_{j}\| \leqslant \|A_{k} \cdot v_{i} - A_{k} \cdot v_{j}\| \leqslant (1 + \varepsilon)\|v_{i} - v_{j}\|$$
Deduce that $\mathbb{P}\left(\bigcap_{1 \leqslant i < j \leqslant N} E_{ij}\right) \geqslant 1 - \frac{N(N-1)}{2}\delta$.