In the book “how to build a brain”, there is this figure about availability of vectors in (high-dimensional) conceptual spaces

However, I could not find an explicit formula or proof for the number of nearly orthogonal vectors available in high-dimensional spaces in the book. I assume, this formula should depend on the dimension $D$ of the vector space and the allowed angle $\epsilon$ between vectors. Furthermore, I assume that it is somehow related to the fact, that the dot-product of two random unit vectors is $\beta$ distributed with zero mean and variance $\sigma = \frac{1}{\sqrt{D}}$. Could you please point me to a paper/reference with an explicit formula (and proof)?