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. 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 the Johnson-Lindenstrauss theorem: there exists an absolute constant $c > 0$ such that for any natural integers $N$ and $d$ greater than or equal to 2 and for any distinct $v_{1}, \ldots, v_{N}$ in $\mathbb{R}^{d}$, it suffices that $$k \geqslant c\frac{\ln(N)}{\varepsilon^{2}}$$ for there to exist an $\varepsilon$-isometry $f : \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$ for $v_{1}, \ldots, v_{N}$.
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. 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 the Johnson-Lindenstrauss theorem: there exists an absolute constant $c > 0$ such that for any natural integers $N$ and $d$ greater than or equal to 2 and for any distinct $v_{1}, \ldots, v_{N}$ in $\mathbb{R}^{d}$, it suffices that
$$k \geqslant c\frac{\ln(N)}{\varepsilon^{2}}$$
for there to exist an $\varepsilon$-isometry $f : \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$ for $v_{1}, \ldots, v_{N}$.