Hermite's constants $\gamma_n$

edit · history · discussion · files · long url · discrete geometry number theory

Numbers

$n$

$\gamma_n$

comment: $\gamma_{1}=1$, $\gamma_{1}^{\,1}=1$; attained by $\mathbb{Z}$
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Z,1,hermite

1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050

comment: $\gamma_{2}=\frac{2}{\sqrt{3}}$, $\gamma_{2}^{\,2}=\frac{4}{3}$; attained by the hexagonal lattice $A_2$ [4]; the same number is in the table of algebraic numbers of degree 2
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,2,hermite

1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094

comment: $\gamma_{3}=2^{1/3}$, $\gamma_{3}^{\,3}=2$; attained by $A_3=D_3$, the face-centred cubic lattice [5]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#A,3,hermite

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: $\gamma_{4}=\sqrt{2}$, $\gamma_{4}^{\,4}=4$; attained by $D_4$ [6]; the same number is in the table of algebraic numbers of degree 2
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,4,hermite

1.515716566510398082347259801306445238681283542978141642037505242097453677202058277641176134849431791

comment: $\gamma_{5}=2^{3/5}$, $\gamma_{5}^{\,5}=8$; attained by $D_5$ [6]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#D,5,hermite

1.665366355311208639217572725017671513324124095787337672980480482451078485985633426184051324797373192

comment: $\gamma_{6}=\frac{2}{3^{1/6}}$, $\gamma_{6}^{\,6}=\frac{64}{3}$; attained by $E_6$ [7]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,6,hermite

1.811447328527813343188345746430206375400891762515874710237416262768844934627125673909528787782071557

comment: $\gamma_{7}=2^{6/7}$, $\gamma_{7}^{\,7}=64$; attained by $E_7$ [7]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,7,hermite

comment: $\gamma_{8}=2$, $\gamma_{8}^{\,8}=256$; attained by $E_8$ [7]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#E,8,hermite

24:

comment: $\gamma_{24}=4$, $\gamma_{24}^{\,24}=4^{24}$; attained by the Leech lattice $\Lambda_{24}$ [10]
equals: Packing_densities_and_Hermite_numbers_of_the_classical_lattices#Lambda,24,hermite

Definition

Hermite's constant $\gamma_n$ [13] is the supremum of the Hermite number $\gamma(L)=\mu(L)/(\det L)^{1/n}$ over all lattices $L\subset\mathbb{R}^n$, where $\mu(L)$ is the squared length of a shortest nonzero vector of $L$ and $\det L$ is the determinant of a Gram matrix of $L$.

Parameters

$n$

— dimension ($n\geq 1$)

Formulas

(1)

$\gamma_n^{\,n}=1,\ \frac{4}{3},\ 2,\ 4,\ 8,\ \frac{64}{3},\ 64,\ 256$ for $n=1,\ldots,8$ [15] [16] and $\gamma_{24}^{\,24}=4^{24}$, so $\gamma_n=1,\ \frac{2}{\sqrt{3}},\ 2^{1/3},\ \sqrt{2},\ 2^{3/5},\ \frac{2}{3^{1/6}},\ 2^{6/7},\ 2$ for $n=1,\ldots,8$ and $\gamma_{24}=4$.

(2)

$\gamma_n=\max_L\gamma(L)$ with $\gamma(L)=\mu(L)/(\det L)^{1/n}$, so $\gamma_n^{\,n}=\mu(L)^n/\det L$ for a lattice $L$ attaining it: $\gamma_n^{\,n}=2^n/\det L$ for the root lattices $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$, $E_8$, all of minimal norm $2$, with $\det A_n=n+1$, $\det D_n=4$, $\det E_6=3$, $\det E_7=2$ and $\det E_8=1$, and $\gamma_{24}^{\,24}=4^{24}$ for the Leech lattice, of minimal norm $4$ and determinant $1$.

(3)

$\gamma_n=4\,\delta_n^{2/n}=4\,(\Delta_n/V_n)^{2/n}$, where $\Delta_n$ and $\delta_n$ are the density and the centre density of the densest lattice sphere packing in $\mathbb{R}^n$ and $V_n=\pi^{n/2}/\Gamma(n/2+1)$ is the volume of the unit ball. The table of the densest known lattice packings holds $\Delta_n$ and $\delta_n$ for $n\leq 8$ and $n=24$; in every other dimension its densities are those of the densest lattice known, and its $4\,\delta_n^{2/n}$ is a lower bound for $\gamma_n$.

(4)

Hermite's inequality $\gamma_n\leq\left(\frac{4}{3}\right)^{(n-1)/2}$ [1], an equality for $n=1,2$; Minkowski's bound $\gamma_n\leq 4\,V_n^{-2/n}$, since $\Delta_n\leq 1$, an equality for $n=1$; Blichfeldt's bound $\gamma_n\leq\frac{2}{\pi}\,\Gamma\!\left(2+\frac{n}{2}\right)^{2/n}$ [8]; and Mordell's inequality $\gamma_n^{\,n-2}\leq\gamma_{n-1}^{\,n-1}$ for $n\geq 3$ [9], an equality for $n=4$ and $n=8$.

Comments

(5)

$\det L$ is the square of the covolume of $L$, so $\sqrt{\gamma_n}$ is the greatest length a shortest nonzero vector of a lattice of covolume $1$ in $\mathbb{R}^n$ can have; in the language of quadratic forms, $\gamma_n$ is the least constant such that every positive definite quadratic form $\sum_{i,j}a_{ij}x_ix_j$ in $n$ variables with determinant $d=\det(a_{ij})$ takes a value at most $\gamma_n d^{1/n}$ at some nonzero integer point [2]. The supremum is attained [2]. Some authors call $\sqrt{\gamma_n}$ Hermite's constant, the ratio of lengths rather than of norms [13], and $\gamma(L)$ is also called the Hermite invariant of $L$; the Hermite numbers of $\mathbb{Z}^n$, the root lattices, their duals, the laminated lattices and the Coxeter–Todd lattice are in the table of the classical lattices, whose notation this table follows.

(6)

$\gamma_n$ is known for $n\leq 8$ and $n=24$ and in no other dimension [13]: by Lagrange for $n=2$ [4], Gauss for $n=3$ [5], Korkine and Zolotareff for $n=4,5$ [6], Blichfeldt for $n=6,7,8$ [7] and Cohn and Kumar for $n=24$ [10], attained by $\mathbb{Z}$, $A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $E_7$, $E_8$ and the Leech lattice $\Lambda_{24}$. In every other dimension the best lower bound known is the Hermite number $4\,\delta_n^{2/n}$ of the densest lattice known, whose centre density $\delta_n$ is in the table of the densest known lattice packings for $n\leq 48$, and upper bounds come from the linear programming bound of Cohn and Elkies [11]. $\gamma_n$ grows linearly in $n$: $\frac{1}{2\pi e}\leq\frac{\gamma_n}{n}\leq\frac{1.744\ldots}{2\pi e}$ for all sufficiently large $n$ [14], the lower bound from Minkowski and Hlawka and the upper bound from Kabatiansky and Levenshtein [12].

(7)

The comment on each entry gives $\gamma_n$ in closed form, the rational number $\gamma_n^{\,n}$ and the lattice attaining $\gamma_n$, and every entry links the row of the table of the classical lattices holding the Hermite number of that lattice.

Programs

(P1)

Sage

hermite = {1: 1, 2: 4/3, 3: 2, 4: 4, 5: 8, 6: 64/3, 7: 64, 8: 256, 24: 4^24}   # gamma_n^n
{n: N(g^(1/n), digits=30) for n, g in hermite.items()}                        # gamma_n
G = CartanMatrix(['E', 6]); n = 6
ZZ(IntegralLattice(G).minimum())^n / G.det()                                   # 64/3 = gamma_6^6

References

[1]

C. Hermite, Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objets de la théorie des nombres, Journal für die reine und angewandte Mathematik 40 (1850), 261–315.

[2]

J. W. S. Cassels, An Introduction to the Geometry of Numbers, Classics in Mathematics, Springer, 1997 (reprint of the 1971 edition), chapter II.

[3]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der mathematischen Wissenschaften 290, Springer, 1999, chapter 1, section 1.5.

[4]

J. L. Lagrange, Recherches d'arithmétique, Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin (1773), 265–312.

[5]

C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen (1831); Werke II, 188–196.

[6]

A. Korkine and G. Zolotareff, Sur les formes quadratiques positives quaternaires, Mathematische Annalen 5 (1872), 581–583; Sur les formes quadratiques positives, Mathematische Annalen 11 (1877), 242–292.

[7]

H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Mathematische Zeitschrift 39 (1935), 1–15.

[8]

H. F. Blichfeldt, The minimum value of quadratic forms, and the closest packing of spheres, Mathematische Annalen 101 (1929), 605–608.

[9]

L. J. Mordell, Observation on the minimum of a positive quadratic form in eight variables, Journal of the London Mathematical Society 19 (1944), 3–6.

[10]

H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Annals of Mathematics 170 (2009), 1003–1050.

[11]

H. Cohn and N. Elkies, New upper bounds on sphere packings I, Annals of Mathematics 157 (2003), 689–714.

[12]

G. A. Kabatiansky and V. I. Levenshtein, On bounds for packings on a sphere and in space, Problems of Information Transmission 14 (1978), 1–17.

Links

[13]

Wikipedia: Hermite constant

[14]

MathWorld: Hermite Constants

[15]

OEIS A007361: numerator of $\gamma_n^{\,n}$

[16]

OEIS A007362: denominator of $\gamma_n^{\,n}$

[17]

OEIS A246184: decimal expansion of $\gamma_6=2/3^{1/6}$

[18]

OEIS A246722: decimal expansion of $\gamma_7=2^{6/7}$

Similar tables

Packing densities and Hermite numbers of the classical lattices — the Hermite numbers $\gamma(L)$ of the classical lattices, among them the nine attaining $\gamma_n$, which every entry here links

Densities of the densest known lattice sphere packings — $4\,\delta_n^{2/n}$ is $\gamma_n$ where the record is proven and the best lower bound known for $\gamma_n$ elsewhere, $n\leq 48$

Volume of the $d$-dimensional unit ball — $V_n$ in $\gamma_n=4\,(\Delta_n/V_n)^{2/n}$ and in Minkowski's bound $\gamma_n\leq 4\,V_n^{-2/n}$

Algebraic numbers of degree 2 — holds $\gamma_2=2/\sqrt{3}$ and $\gamma_4=\sqrt{2}$, which the entries for $n=2$ and $n=4$ link

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (every dimension in which $\gamma_n$ is known is here, $1\leq n\leq 8$ and $n=24$)

How they were obtained:

The datum for each entry is the rational number $\gamma_n^{\,n}$, a theorem in each dimension listed; $\gamma_n$ is written exactly where that rational is an $n$-th power ($n=1$, $8$, $24$) and otherwise as the ball for its $n$-th root in arb, computed with 64 guard bits beyond the 100 digits written; the widest ball relative to its value, $\gamma_7$, has radius $1.4\cdot 10^{-119}$.

Before a value is returned the generator recomputes $\gamma_n^{\,n}$ for $n\leq 8$ as $\mu^n/\det L$ from the Gram matrix of the attaining lattice, the identity matrix or the Cartan matrix, with $\mu$ from PARI's qfminim. Before any entry was written, $\gamma_n^{\,n}$ was compared with OEIS A007361/A007362 for $n\leq 8$; $\gamma_2$ to $\gamma_7$ with OEIS A020832 (ten times $\gamma_2$), A002580, A002193, A011093, A246184 and A246722 to every digit those entries give; the nine closed forms with those of the Wikipedia and MathWorld pages, evaluated in balls; $\gamma_n$ with the largest stored Hermite number in dimension $n$ of the table of the classical lattices, whose linked row holds the same hundred digits, and with $4\,\delta_n^{2/n}$ from the stored centre densities of the table of the densest known lattice packings; $\gamma_{24}^{\,24}$ with $\mu^{24}/\det$ of the Leech lattice's Gram matrix from the Catalogue of Lattices, with $\mu=4$ and $196560$ minimal vectors from qfminim; and Hermite's, Minkowski's, Blichfeldt's and Mordell's inequalities, with the stated equality cases, on every entry, Minkowski's against the stored digits of the table of unit-ball volumes as well. Controls that must fail did: $\gamma_4$ moved by $10^{-50}$ no longer matches A002193, and $\Lambda_{23}$ offered as the lattice attaining $\gamma_{24}$ is refused.