Let $A_d(n)$ be the largest number of points that can be packed on the $n$-unit sphere, such that every point is at least $d$ apart. Compare with, for instance, https://arxiv.org/abs/1507.03631
When d=1, these are the standard kissing numbers, and they are known to grow exponentially in n. For any smaller d we can see that it must be exponential as well, because there is a simple exponential upper bound based on solid angles subtended by $A$-many circles on the surface.
For d=√2, these are vectors that are "at least" orthogonal (nonpositive inner product), and $A_d(n)$ is just 2n, in an octahedral packing. This implies that for any d>√2, the growth in n is at most linear.
On a smaller note, for d≥√3, A_d(n) is a constant in n.
Question: What happens for values of $d \in (1,\sqrt 2)$? Is it true that $A_d(n)$ is always either asymptotically linear, or at least exponential? If so, is the transition at d=√2 exactly?
And a side question, is $A_d(n)$ always either asymptotically linear or constant on $d\in(\sqrt 2,\sqrt 3)$?