Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ denote the Euclidean unit sphere in $\mathbb{R}^d$. For constants $\alpha, \beta$ such that $[\alpha, \beta]$ is a subinterval of $[-1, 1]$, define

$$\mathcal{P}_{[\alpha, \beta]}(d) = \{S \subseteq \mathcal{S}^{d-1}: \text{for all distinct } \mathbf{x}, \mathbf{y} \in S \text{ it holds that } \alpha \leq \langle \mathbf{x}, \mathbf{y}\rangle \leq \beta\}.$$

In other words, dot products between any two vectors in $S$ must always lie in the interval $[\alpha, \beta] \subseteq [-1, 1]$ for $S$ to be contained in $\mathcal{P}_{[\alpha, \beta]}$. For certain values of $\alpha$ and $\beta$ I am interested in (upper bounds on) high-dimensional asymptotics of the maximal size of such sets $S \in \mathcal{P}_{[\alpha, \beta]}(d)$:

$$M_{[\alpha, \beta]} = \lim_{d \to \infty} \frac{1}{d} \log_2\left(\max_{S \in \mathcal{P}_{[\alpha, \beta]}(d)} |S|\right)$$

Without the condition on the minimum dot product (i.e. with $\alpha = -1$), we are looking for standard sphere packing bounds, and e.g. the famous Kabatiansky-Levenshtein result states $M_{[-1, 1/2]} \leq 0.4014135\dots$. For $\alpha > -1$, the additional condition can be interpreted as not only putting spheres around points $\mathbf{x} \in S$, but also putting spheres of potentially different radius around $-\mathbf{x}$.

For $\alpha = -\beta$, the additional condition with $\alpha$ can be interpreted as only considering sphere packings where either none or both of $\pm\mathbf{x}$ are included in packings $S$, for arbitrary $\mathbf{x} \in \mathcal{S}^{d-1}$. Since any packing in $\mathcal{P}_{[-\beta, \beta]}$ induces a packing in $\mathcal{P}_{[-1, \beta]}$, one trivially has $M_{[-\beta, \beta]} \leq M_{[-1, \beta]}$, which together with Kabatiansky-Levenshtein's results leads to upper bounds on $M_{[-\beta, \beta]}$.

Note that the conditions on spheres around $\mathbf{x}$ and $-\mathbf{x}$ are not quite symmetric: it is possible to construct arbitrarily large sets $S \in \mathcal{P}_{[\alpha, 1]}$ by clustering a large number of points around an arbitrary point $\mathbf{x} \in \mathcal{S}^{d-1}$. In a sense, the upper bound $\beta$ on the inner products is more restrictive than the lower bound $\alpha$.

Some concrete research questions that came up:

  1. What can be said about $M_{[-\beta, \beta]}$? Is it the same as $M_{[-1, \beta]}$? In other words: is the maximum packing density of antipodal packings the same as of arbitrary sphere packings?

  2. For $|\alpha| < \beta$, what can be said about $M_{[\alpha, \beta]}$ other than $M_{[\alpha, -\alpha]} \leq M_{[\alpha, \beta]} \leq M_{[-\beta, \beta]}$? Can we perhaps show that $M_{[\alpha, \beta]}$ is the same as $M_{[\alpha, -\alpha]}$ if $\beta$ is not too close to $1$?

If general remarks about $M_{[\alpha, \beta]}$ are too much to ask for, I'm mostly interested in finding non-trivial upper bounds for $M_{[-1/k, 1/2]}$ for integer $k \geq 2$.

Any help or pointers to relevant literature would be greatly appreciated!

Edit: To add, Kabatiansky and Levenshtein studied the first problem above as well, and concluded that their method does not give sharper upper bounds on $M_{[-\beta,\beta]}$ than those obtained by first using the trivial relation $M_{[-\beta,\beta]} \leq M_{[-1,\beta]}$ and then applying their upper bound on $M_{[-1,\beta]}$. They conjecture that $M_{[-1,\beta]} = M_{[-\beta, \beta]}$ just after Equation (69) in the English version of the paper, but they do not give a proof. Alas, it seems that to obtain sharper bounds on $M_{[-1,\beta]}$ (or proving that there do not exist sharper bounds) it does not suffice to only use K-L's techniques.

  • $\begingroup$ Anyone? (Is this likely an open problem that cannot be solved easily?) $\endgroup$ – TMM Feb 2 '18 at 18:43

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