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.

  • 1
    $\begingroup$ You might be able to get something useful from results on Riesz energy minimizing point configurations e.g. arxiv.org/abs/1202.4037 in the limit $s\to\infty$. $\endgroup$ Nov 26, 2019 at 19:46
  • $\begingroup$ If you have access to MathSciNet, that might be an improvement on Google. $\endgroup$ Nov 26, 2019 at 21:00
  • $\begingroup$ @Yoav Kallus: Maybe this warning from the hlinked paper, that Riesz s-energy is a red herring: "Since for s ≥ d the s-energy integral for every positive Borel probability measure supported on $S^d$ is +∞, potential theory fails to work.". Does it qualify as useful? $\endgroup$ Nov 28, 2019 at 11:49

1 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.

  • $\begingroup$ That paper indeed; 1951 not 1955, my mistake. I believe they provide a lower bound, not an upper bound, for $M$ as a function of $a$. Their formula (17) maximizes $a$ for a fixed $M$, in my notations; two quantities they named $1/R$ and $N$ respectively. $\endgroup$ Nov 27, 2019 at 13:36
  • $\begingroup$ Yes, this difference between the two papers was overlooked. It is fixed above now. $\endgroup$ Nov 27, 2019 at 14:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.