The Sharp Dimension Bound in the Johnson--Lindenstrauss Lemma

The Johnson–Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O({\epsilon }^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\epsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d,\epsilon$ is $$\mathrm{\Theta }(\min \{d,n-1,\frac{\log (2+{\epsilon }^{2}n)}{{\epsilon }^2}\})\mathrm{.}$$ We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen–Nelson and Alon–Klartag, holds even for nonlinear embeddings.