Superposition is the possibility of storing more than $n$ features in an $n$-dimensional vector, by letting the features be slightly correlated with each other. It turns out that one can store exponentially many features in a given vector. The ability to store that many features in a single vector space is sometimes explained using the Johnson–Lindenstrauss lemma, but the lemma seems counterintuitive, so I came up with an alternative approach that I found simpler:

Suppose you have a set $F$ with $2^d$ elements and you want to embed it in a $d$-dimensional vector space. We label each element $x_i$ with integers $i$ such that $F=\{x_0,x_1,\dots ,x_{2^d-1}\}$. You can write each integer $i$ as a string of $d$ bits, $b_{d-1}\dots b_2{b}_1{b}_0$. To improve symmetry, we translate a bit $b$ from being $0$ or $1$ to being $-1$ or $1$ by taking $2b-1$. Join all the digits into a vector and normalize to get the embedding:

$e\left(x_i\right)=\frac{1}{\sqrt{d}}[2b_{d-1}-1,\dots ,2b_2-1,2b_1-1,2b_0-1{]}^{\top }$

That is, we map each bit to a separate dimension, with a $b=1$ bit mapping to a positive value and a $b=0$ bit mapping to a negative value, and scale the embedding by $\frac{1}{\sqrt{d}}$ to keep the embedding vector of unit length.

If we pick two random elements $x_a$ and $x_b$ of $F$, then an elementary argument shows that their dot product is well-approximated as following a normal distribution $N(0,\frac{1}{\sqrt{d}})$.

In some ways this isn't quite as perfect as the Johnson-Lindenstrauss lemma since you could in principle be unlucky and get two elements that accidentally have a high similarity. After all, for a given element $x_i$, there will be $d$ elements $x_{i\oplus 2^k}$ whose numbers merely differ from $x_i$ by a bitflip. However, it is straightforward to reduce the noise: just concatenate multiple embeddings based on different labels. If instead of using $d$ dimensions, you use $Rd$ dimensions, then you can pump down the noise to $N(0,\frac{1}{\sqrt{Rd}})$.