Exponential improvements for superball packing upper bounds

TL;DR

This paper establishes the first exponential improvement in upper bounds for the packing density of unit b5c1-balls in high-dimensional spaces for all fixed p > 2, advancing understanding of geometric packing limits.

Contribution

It proves a new exponential upper bound on translative packing density of b5c1-balls in b5c4-dimensional space for all p > 2, improving previous bounds since 1936.

Findings

New exponential upper bounds for packing densities

First such improvements since 1936

Bound b3_p < -1/p for all p > 2

Abstract

We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since van der Corput and Schaake (1936).

Figures

Equations

$${\gamma }_p:=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}{\log }_2{\Delta }_p(n)$$

$${\Delta }_2(n)\le 2^{({\kappa }_{\text{KL}}+o(1))n},\text{where }{\kappa }_{\text{KL}}:=-0.5990\dots .$$

$${\gamma }_p\le -1/p\text{for }p>2.$$

$${\gamma }_p\le \underset{0<\theta <\pi /2}{\inf }\left(\frac{1+\sin \theta }{2\sin \theta }{\log }_2\frac{1+\sin \theta }{2\sin \theta }-\frac{1-\sin \theta }{2\sin \theta }{\log }_2\frac{1-\sin \theta }{2\sin \theta }+\frac{2}{p}{\log }_2\sin \frac{\theta }{2}\right).$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}{\log }_2{A}_2(n,\sin (\theta /2))\le a(\theta )$$

$$a(\theta ):=\frac{1+\sin \theta }{2\sin \theta }{\log }_2\frac{1+\sin \theta }{2\sin \theta }-\frac{1-\sin \theta }{2\sin \theta }{\log }_2\frac{1-\sin \theta }{2\sin \theta }.$$

$${\Delta }_2(n)\le {\sin }^n(\theta /2)A_2(n+1,\sin (\theta /2)),$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}{\log }_2{\Delta }_2(n)\le a(\theta )+{\log }_2\sin \frac{\theta }{2}.$$

$$x^{\ast }:=\mathrm{sgn}(x)|x{|}^{p/2},x\in \mathbb{R}.$$

$$|x^{\ast }-y^{\ast }|\ge 2^{1-p/2}|x-y{|}^{p/2}.$$

$$x^{p/2}-y^{p/2}\ge (x-y{)}^{p/2}\ge 2^{1-p/2}(x-y{)}^{p/2}.$$

$${\Vert {\mathbf{x}}^{\ast }-{\mathbf{y}}^{\ast }\Vert }_2^2=\sum _{i=1}^n|x_i^{\ast }-y_i^{\ast }{|}^2\ge 2^{2-p}\sum _{i=1}^n|x_i-y_i{|}^p\ge 2^2{d}^p,$$

$${\Delta }_p(n)\le \sin (\theta /2{)}^{2n/p}{A}_p(n+1,\sin (\theta /2{)}^{2/p})\le \sin (\theta /2{)}^{2n/p}{A}_2(n+1,\sin (\theta /2)).$$

$${\gamma }_p=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}{\log }_2{\Delta }_p(n)\le a(\theta )+\frac{2}{p}{\log }_2\sin (\theta /2).$$

$${\gamma }_p\le -\frac{1}{p}-\frac{\eta }{p\ln 2}+o(\eta ).$$

$${\gamma }_p\le -\frac{1}{p}-\frac{1}{\ln 4}\cdot \frac{1}{p^2\ln p}+O\left(\frac{1}{p^2{\ln }^{2}p}\right),\text{as }p\to \infty .$$

$${\gamma }_p\le {\kappa }_{\text{KL}}+\frac{2-p}{p}{\log }_2\sin \frac{{\theta }_{KL}}{2},\text{for all }p\ge 2.$$

$${\gamma }_{2+ϵ}\le {\gamma }_{\text{KL}}-\left(\frac{1}{2}{\log }_2\sin \frac{{\theta }_{\text{KL}}}{2}\right)ϵ+O({ϵ}^2)=(-0.5990\dots )+(0.4650\dots )ϵ+O({ϵ}^2).$$

$${\gamma }_p\ge -(1+o(1))\frac{\ln \ln p}{p\ln 2},\text{as }p\to \infty ,$$

$${\gamma }_{2+ϵ}\ge -1+\left(\frac{\sqrt{\pi }\zeta (3)}{2\ln 2}+o(1)\right)\frac{ϵ}{{\ln }^{3/2}(1/ϵ)},\text{as }ϵ\to 0^{+}.$$

$${\gamma }_p\le \underset{\frac{1}{2}\le \frac{1}{q}\le \frac{1}{3}\left(1+\frac{1}{p}\right)}{\inf }\left(b(p)-b(q)-1+1/p+(1/q-1/p){\log }_2\left(\frac{2-1/q}{1-1/q}\right)\right)$$

$$b(p):=\underset{n\to \infty }{\lim }\frac{1}{n}{\log }_2\mathrm{vol}{\mathbf{B}}_p^n(n^{1/p})=1+{\log }_2\Gamma \left(1+\frac{1}{p}\right)+\frac{1}{p}{\log }_2(pe).$$

$$\frac{{\Delta }_p(n)}{\mathrm{vol}{\mathbf{B}}_p^n(n^{1/p})}\le \frac{{\Delta }_q(n)}{\mathrm{vol}{\mathbf{B}}_p^n(n^{1/q})}.$$

$${\gamma }_p\le {\kappa }_{\text{KL}}-b(2)+b(p)=(-0.5990\dots )-b(2)+b(p)\text{for }1\le p<2.$$

$${\gamma }_p\le \min \{\text{RHS of eqrefeq:RankinIII},\text{RHS of eqrefeq:p<2-KL}\}$$

Full text

YZ was supported by NSF Awards DMS-1362326 and DMS-1764176, and the MIT Solomon Buchsbaum Fund.

1. Introduction

The sphere packing problem asks for the densest packing of non-overlapping unit balls in $\mathbb{R}^{n}$. This is an old and difficult problem whose exact solution is only known in dimensions 1, 2, 3, 8, and 24. The problem is already non-trivial in two dimensions (see [8] for a short proof). The three-dimensional sphere packing problem is known as Kepler’s conjecture, and it was solved by Hales [9] via a monumental computer-assisted proof. The problem in eight dimensions was recently resolved by Viazovska [23] in a stunning breakthrough, and the method was then quickly extended to solve the problem in twenty-four dimensions [3]. Dimensions 8 and 24 are special due to the existence of highly dense and symmetric lattices known as the $E_{8}$ lattice (dimension 8) and the Leech lattice (dimension 24). See the survey [2] and its references for background and recent developments.

In this paper, we study translative packings of $\ell_{p}$-balls in high dimensions. Denote the $\ell_{p}$-balls with radius $R$ in $\mathbb{R}^{n}$ by $\bm{B}_{p}^{n}(R):=\{\bm{x}\in\mathbb{R}^{n}:\left\lVert\bm{x}\right\rVert_{p}\leq R\}$ and the unit $\ell_{p}$-ball by $\bm{B}_{p}^{n}:=\bm{B}_{p}^{n}(1)$. Here $\left\lVert(x_{1},\dots,x_{n})\right\rVert_{p}:=(|x_{1}|^{p}+\cdots+|x_{n}|^{p})^{1/p}$ is the $\ell_{p}$-norm. The name superball refers to $\ell_{p}$-balls with $p>2$ [19]. Superballs are more cube-like compared to the familiar $\ell_{2}$-balls. See [10, 11, 5] for studies of $\ell_{p}$-ball packings in $\mathbb{R}^{3}$. Although $\ell_{p}$-balls do not possess rotational symmetry, in this paper we only consider translations of identical $\ell_{p}$-balls, not allowing rotations. The best known lower bounds on high dimensional superball packing densities do not use rotations [6] (see Section 3.2).

Let $\Delta_{p}(n)$ denote the maximum translative packing density of copies of $\bm{B}_{p}^{n}$ in $\mathbb{R}^{n}$. Here density is the fraction of space occupied by these balls. For fixed $p\in[1,\infty)$, let

[TABLE]

be the exponential rate of optimal packing densities in high dimensions. The precise value of $\gamma_{p}$ is unknown for any $p\in[1,\infty)$, and the current best upper and lower bounds are quite far apart. For Euclidean balls, $p=2$, the best high dimensional upper bound (apart from constant factors) is due to Kabatiansky and Levenshtein [12]:

[TABLE]

See Cohn and Zhao [4] and Sardari and Zargar [20] for constant factor improvements over Kabatiansky and Levenshtein [12]. For lower bounds, we have $\Delta_{p}(n)\geq 2^{-n}$ for all $n$ and $p\geq 1$ since every maximal packing has density at least $2^{-n}$. For $p=2$, there have only been subexponential improvements, with the current best lower bound due to Venkatesh [22]. In summary, the best bounds on $\gamma_{2}$ are $-1\leq\gamma_{2}\leq\kappa_{\textrm{KL}}=-0.5990\dots$.

For $p>2$, the current best upper bound on the exponential rate of superball packing densities was first proved by van der Corput and Schaake [21] via Blichfeldt’s method [1] (e.g., see [24, Section 6.3]), giving

[TABLE]

There have been subsequent subexponential upper bound improvements on $\Delta_{p}(n)$ for $p>2$, e.g., Rankin [16, 17]. We defer to Section 3 for a discussion of known bounds on $\gamma_{p}$ in other regimes.

In this paper, we prove a new upper bound on $\gamma_{p}$ for all $p>2$, giving the first exponential improvement since 1936 on the upper bound of superball packing densities in high dimensions.

Theorem 1.1**.**

For all $p\geq 2$,

[TABLE]

In particular, $\gamma_{p}<-1/p$ for all $p\geq 2$.

See Figure 1 for a plot of the bounds.

Remark*.*

Theorem 1.1 with $p=2$ recovers $\gamma_{p}\leq\kappa_{\textrm{KL}}$. Our upper bound on $\gamma_{p}$ is continuous with $p$, whereas the previous best bounds were not continuous111It is unknown whether $p\mapsto\gamma_{p}$ is continuous. Lemma 3.1 implies that $\gamma_{p}$ is continuous at all but at most countably many points. at $p=2$. The fact that our bound at $p=2$ recovers the Kabatiansky–Levenshtein bound is not a coincidence, as our proof relies on the Kabatiansky–Levenshtein bound for spherical codes.

2. Proof of main theorem

2.1. Kabatiansky–Levenshtein spherical code bound

Denote the $\ell_{p}$-sphere in $\mathbb{R}^{n}$ of radius $R$ by $\bm{S}^{n-1}_{p}(R):=\{\bm{x}\in\mathbb{R}^{n}:\left\lVert\bm{x}\right\rVert_{p}=R\}$ and the unit $\ell_{p}$-sphere by $\bm{S}_{p}^{n-1}:=\bm{S}_{p}^{n-1}(1)$. Let $A_{p}(n,d)$ to be the maximum number of points on $\bm{S}_{p}^{n-1}$ with pairwise $\ell_{p}$-distance at least $2d$, i.e, an $\ell_{p}$-spherical code. Note that $A_{p}(n,d)=1$ unless $d\in[0,1]$. Note that $A_{2}(n,\sin(\theta/2))$ is the maximum size of a spherical code in $\mathbb{R}^{n}$ with pairwise angle at least $\theta$. Kabatiansky and Levenshtein [12] proved that for all222A simple geometric argument (see [13, (17)]) shows that the upper bound (2.1) can be improved for $\theta<\theta_{\textrm{KL}}:=1.0995\dots$ to $a(\theta_{\textrm{KL}})+\log_{2}\sin(\theta_{\textrm{KL}}/2)-\log_{2}\sin(\theta/2)$, but this improvement does not benefit our bounds. $0<\theta<\pi/2$,

[TABLE]

where

[TABLE]

A projection argument (see [4, Section 2]) shows that

[TABLE]

so (2.1) gives

[TABLE]

The bound $\gamma_{2}\leq\kappa_{\textrm{KL}}=-0.5990\dots$ is obtained by choosing $\theta=\theta_{\textrm{KL}}=1.0995\dots$ to minimize the upper bound above.

2.2. $\ell_{p}$-twist

Fix $p\geq 2$. Define

[TABLE]

For $\bm{x}=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$, write $\bm{x}^{*}:=(x_{1}^{*},\dots,x_{n}^{*})$, and for $X\subseteq\mathbb{R}^{n}$, write $X^{*}:=\{\bm{x}^{*}:\bm{x}\in X\}$.

Observe that for all $x,y\in\mathbb{R}$,

[TABLE]

Indeed, without loss of generality it suffices to consider two cases: $x\geq 0\geq y$ and $x\geq y\geq 0$. The former case is an immediate consequence of Hölder’s inequality (or the convexity of $x\mapsto x^{p/2}$). In the latter case, we have

[TABLE]

Here we use $(w+z)^{p/2}\geq w^{p/2}+z^{p/2}$ for $w,z>0$, which can be proved by first normalizing to $w+z=1$ and noting that $w^{p/2}+z^{p/2}\leq w+z=1$.

Lemma 2.1**.**

For all $p\geq 2$ and $d\in(0,1]$, we have $A_{p}(n,d)\leq A_{2}(n,d^{p/2})$.

Proof.

Let $X\subseteq\bm{S}_{p}^{n-1}$ with $|X|=A_{p}(n,d)$ and $\left\lVert\bm{x}-\bm{y}\right\rVert_{p}\geq 2d$ for all distinct $\bm{x},\bm{y}\in X$. We have $\left\lVert\bm{x}^{*}\right\rVert_{2}=\left\lVert\bm{x}\right\rVert_{p}=1$ for all $\bm{x}\in X$, so $X^{*}\subseteq\bm{S}_{2}^{n-1}$. For distinct $\bm{x},\bm{y}\in X$, we have

[TABLE]

by (2.2). Thus $X^{*}$ is a subset of $\bm{S}_{2}^{n-1}$ whose points have pairwise $\ell_{2}$-distance at least $2d^{p/2}$. Hence $|X|=|X^{*}|\leq A_{2}(n,d^{p/2})$. ∎

Remark*.*

The same argument shows that $A_{p}(n,d)\leq A_{q}(n,d^{p/q})$ for all $1\leq q\leq p$ and $d\in(0,1]$.

Lemma 2.2**.**

For every $p\geq 1$, $d\in(0,1]$, and $n\in\mathbb{N}$, we have $\Delta_{p}(n)\leq d^{n}A_{p}\left(n+1,d\right)$.

Proof.

Let $\rho<\Delta_{p}(n)$ be arbitrary. Consider a translative packing $\{\bm{x}+\bm{B}_{p}^{n}(d):\bm{x}\in X\}$ in $\mathbb{R}^{n}$ with density greater than $\rho$, where $X\subseteq\mathbb{R}^{n}$ is the set of centers of the $\ell_{p}$-balls. By an averaging argument333A uniform random translation of a unit $\ell_{p}$-ball inside $[-R,R]^{n}$ contains more than $d^{-n}\rho+o_{R\to\infty}(1)$ points of $X$., there exists some translate of a unit $\ell_{p}$-ball that contains at least $d^{-n}\rho$ points of $X$. Translating $X$ if necessary, we may assume that $|X\cap\bm{B}_{p}^{n}|\geq d^{-n}\rho$. Add an $(n+1)$-st coordinate to each point in $X\cap\bm{B}_{p}^{n}$ to obtain a set $X^{\prime}$ of points on the unit $\ell_{p}$-sphere in $\mathbb{R}^{n+1}$. In other words, $X^{\prime}$ is obtained by projecting the points of $X$ contained in the unit ball “upward” to the hemisphere one dimension higher. Since the points in $X$ are pairwise at least $2d$ apart in $\ell_{p}$-distance, the same holds for $X^{\prime}$. So $X^{\prime}$ is an $\ell_{p}$-spherical code whose points are pairwise separated by $\ell_{p}$-distance at least $2d$, and hence $d^{-n}\rho\leq|X\cap\bm{B}_{p}^{n}|=|X^{\prime}|\leq A_{p}(n+1,d)$. Since $\rho$ can be arbitrarily close to $\Delta_{p}(n)$, we obtain the claimed inequality. ∎

Remark*.*

As in [4], the above argument can be modified so that we do not need to add a new dimension when $d\in[1/2,1]$, resulting in a slightly better bound $\Delta_{p}(n)\leq d^{-n}A_{p}(n,d)$. We omit the details of this modification since this improvement does not affect the exponential asymptotics.

Proof of Theorem 1.1.

Applying Lemmas 2.2 and 2.1, we have, for every $0<\theta<\pi/2$,

[TABLE]

Applying (2.1), we obtain

[TABLE]

The main result follows by taking the infimum of the bound over $\theta\in(0,\pi/2)$.

Setting $\theta=\pi/2-\eta$, we have, with $p\geq 2$ fixed and $\eta\to 0^{+}$,

[TABLE]

So choosing $\eta>0$ sufficiently small gives $\gamma_{p}<-1/p$ for all $p\geq 2$. ∎

3. Remarks

3.1. Asymptotics

Setting $\theta=\pi/2-(p\ln p)^{-1}$, we obtain

[TABLE]

Taking $\theta=\theta_{\textrm{KL}}$ gives

[TABLE]

Thus, as $\epsilon\to 0^{+}$,

[TABLE]

3.2. Review of other bounds on $\gamma_{p}$

Here we survey other existing bounds on $\gamma_{p}$.

For $p=2$, the best known bounds are $-1\leq\gamma_{2}\leq\kappa_{\textrm{KL}}=-0.5990\dots$ as discussed earlier.

For $p>2$, the best known upper bounds are the ones given in this paper. For lower bounds, extending on methods developed by Rush [18] and Rush–Sloane [19], Elkies, Odlyzko, and Rush [6] proved $\gamma_{p}>-1$ for all $p>2$, thereby exponentially beating the Minkowski–Hlawka lower bound. See [6] for the precise bound. Their bounds have the following asymptotics:

[TABLE]

and

[TABLE]

Here $\zeta$ denotes the Riemann zeta function. See [14] for some later improvements using algebraic-geometric codes for some specific integers $p$.

For $1\leq p<2$, no improvement over the Minkowski–Hlawka lower bound $\gamma_{p}\geq-1$ is known. The best upper bound on $\gamma_{p}$ is due to Rankin [15], based on Blichfeldt’s method [1]:

[TABLE]

where

[TABLE]

Recall that $\mathop{\text{vol}}\bm{B}_{p}^{n}=2^{n}\Gamma(1+1/p)^{n}/\Gamma(1+n/p)$.

For packings of congruent cross-polytopes (i.e., unit $\ell_{1}$-balls) allowing rotations, Fejes Tóth, Fodor, and Vígh [7] proved an exponentially decaying upper bound in high dimensions. For translative packing of unit $\ell_{1}$-balls, the upper bound 3.1 remains best known in high dimensions.

We note that the above bound (3.1) can be improved on the region $p\in[1.494\dots,2)$ using the Kabatiansky–Levenshtein bound via the following folklore observation.

Lemma 3.1**.**

For $1\leq p\leq q\leq\infty$, $\gamma_{p}-b(p)\leq\gamma_{q}-b(q)$.

Proof.

By monotonicity of norms, we have $n^{-1/p}\|\bm{x}\|_{p}\leq n^{-1/q}\|\bm{x}\|_{q}$, so $\bm{B}_{p}^{n}(n^{1/p})\supseteq\bm{B}_{q}^{n}(n^{1/q})$. Any packing of $\bm{B}_{p}^{n}(n^{1/p})$ can be shrunk into a packing of $\bm{B}_{q}^{n}(n^{1/q})$. Hence

[TABLE]

Taking log, dividing by $n$, and letting $n\to\infty$ yields the lemma. ∎

Using $\gamma_{2}\leq\kappa_{\textrm{KL}}$, we find that

[TABLE]

Thus

[TABLE]

See Figure 1 for an illustration of the above bounds.

Acknowledgments

This work began during Y.Z.’s internship at Microsoft Research New England, and he would like to thank Henry Cohn for discussions and mentorship and Microsoft Research for its hospitality.