Packing disks of infinitesimal diameter on a sphere: the asymptotics of the Tammes problem

Question

This is an elaboration on MO Question 212550: given $ 0 < 2a << 1 $, how many points can be placed on the unit sphere, subject to the constraint that any two of these points must be at Euclidean distance at least $2a$ ?

Obviously, every point must be allocated at least the area of a disk with radius $a$, so the maximum number $M(a)$ cannot exceed $4/a^2$; however, Fejes Tóth famously proved that the allocated area must be at least that of the circumscribing hexagon, so that: $2 \sqrt 3 a^2 M(a) = 4\pi - S(a)$ with $S(a) > {2\over3}\pi a^2 $.

Then, we have a 1955 (Edit: 1951) paper by Habicht & Van der Waerden offering a very contorted construction, and the estimate: $S(a) = O(a^{2/3}) $. They concluded with an oddly phrased comment, acknowledging their explicit value for the constant is far from minimal, while implying the order of magnitude is tight, i. e. their $O(a^{2/3}) $ should really be $\Theta(a^{2/3}) $.

Then... that's all. Or at least, that's all I could google regarding my question. To wit:

How much is proven, and how much more is plausible, of the behaviour of $S(a)\over a^{2/3}$ for inifinitesimal $ a $? If it is $o(1) $, what is its order of magnitude? If it is $\Theta(1) $, what are its upper & lower limits?

Edit: I do not mean Google has no answer to this question; only that they lack relevance. J. Park pointed out in his answer that any explicit construction provides a lower bound; a plethora of such constructions result in $S(a) < C a^{2/3}$ for various $C$'s, some better, some worse than the original one by H.-v.d.W. What's wanting is a public claim of optimality.

Answer 1

The second-order behavior of $M(a)$ for $\mathbb{S}^2$ was investigated in van der Waerden's paper Punkte auf der Kugel shortly after the first paper by Habicht and van der Waerden was published (Lagerung von Punkten aut der Kugel appears to be the paper you are referring to, although it was published in 1951).

There is only an improved lower bound for $M(a)$ given in the later paper (one needs also to renormalize quantities there to agree with those in this question).

Edit: Note that any construction gives a lower bound on $M(a)$. It is difficult to give asymptotically optimal constructions to the first order (see Chapter 3.3 of Hamkin's thesis here for a review of lower bounds). Fejes Tóth's upper bound gives to first order a comparable bound (for $d=2a$):

$$M(d)\leq 2\left[1-\frac{\pi/6}{\cot^{-1}\sqrt{3-d^2}}\right]^{-1}.$$

It is not clear whether there are matching bounds known to the second order.