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Hao CHEN et Liqing XU
Counterexamples to the Woods Conjecture in dimensions d ≥ 24
Tome 31, no 3 (2019), p. 723-726.
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Journal de Théorie des Nombres
de Bordeaux 31 (2019), 723–726
Counterexamples to the Woods Conjecture in
dimensions d ≥ 24
par Hao CHEN et Liqing XU
Résumé. Soit Nd le maximum des rayons de recouvrement des réseaux d-
dimensionnels unimodulaires possédants d vecteurs minimaux indépendants.
En 1972, A. C. Woods a conjecturé que Nd ≤
√
d
2 . En 2005, C. T. McMul-
len a démontré que la conjecture de Woods implique la célèbre conjecture
de Minkowski. La conjecture de Woods est prouvée pour d ≤ 9. En 2016,
Regev, Shapira et Weiss ont trouvé des contre-exemples à la conjecture de
Woods pour d ≥ 30. Dans cet article, nous donnons des contre-exemples à la
conjecture de Woods pour d ≥ 24. La question reste donc ouverte pour les
dimensions 10 ≤ d ≤ 23.
Abstract. Let Nd be the greatest value of covering radius over all well-
rounded unimodular d dimensional lattices. In 1972 A. C. Woods conjectured
that Nd ≤
√
d
2 . C. T. McMullen proved that the Woods conjecture implies the
celebrated Minkowski’s conjecture in 2005. The Woods conjecture has been
proved for d ≤ 9. In 2016 Regev, Shapira and Weiss gave counterexamples for
the Woods conjecture for d ≥ 30. In this paper we give counterexamples to
the Woods conjecture in dimensions d ≥ 24. Then the unknown dimensions
of the Woods conjecture are 14 dimensions 10 ≤ d ≤ 23.
1. Introduction
A d dimensional lattice L ⊆ Rd is the set of all integer coefficient linear
combinations of d linearly independent vectors b1, . . . ,bd.
L := {x = x1b1 + · · ·+ xdbd : x1 ∈ Z, . . . , xd ∈ Z}.
The volume of this lattice is
vol(L) = |det(b)|
where b is the d× d matrix with d rows b1, . . . ,bd. L is called unimodular
if vol(L) = 1. The length of shortest nonzero lattice vectors is λ1(L) =
minx∈L−0{
√
〈x,x〉}. If lattice vectors with the length λ1(L) in L span Rd,
Manuscrit reçu le 25 janvier 2019, révisé le 15 mars 2019, accepté le 8 avril 2019.
2010 Mathematics Subject Classification. 11H31, 11H99.
Mots-clefs. Lattice, Woods conjecture, Minkowski conjecture.
We are grateful to the referee for their helpful comments. This research was supported by the
NSFC Grant 11531002 and the Major Program Guangdong Basic and Applied Research Grant
2019B030302008.
724 Hao Chen, Liqing Xu
it is called well-rounded. An unimodular lattice L is called critical if the
function λ1(L) attains a local maximum at L. We refer [1] for the detail of
lattice. It was proved by Voronoi that a critical lattice is well-rounded, we
refer this result to [2, Chpt. 6].
The covering radius r(L) = maxy∈Rd,x∈L{
√
〈y− x,y− x〉}. Let
Nd = max
LWR,vol(L)=1
{r(L)}
be the greatest value of covering radius over all well-rounded unimodular d
dimensional lattices. The following conjecture was due to A. C. Woods [8]
and has been proved for d ≤ 9 (see [3]).
Conjecture 1.1 (The Woods conjecture). Nd ≤
√
d
2 .
Let N(y) = |y1 . . . yd| for y = (y1, . . . , yd) ∈ Rd. For a vector a =
(a1, . . . , ad) ∈ Rd, the lattice a · Z := {(a1x1, . . . , adxd) : x1 ∈ Z, . . . , xd ∈
Z}. The following conjecture due to H. Minkowski has been studied for
many years.
Conjecture 1.2 (Minkowski’s conjecture). For any unimodular lattice L
in Rd,
sup
y∈Rd
inf
x∈L
N(y− x) ≤ 2−d.
The equality holds if and only if there exist positive a1, . . . , ad such that
a1a2 . . . ad = 1 and L = a · Zn.
In 2005 McMullen’s breakthrough JAMS paper [4] it was proved that
the Woods conjecture implies Minkowski’s conjecture. Hence Minkowski’s
conjecture is true for d ≤ 9 (see [3]). Unfortunately in 2016, counterexam-
ples to the Woods conjecture was given for d ≥ 30 by Regev, Shapira and
Weiss in [5] and a lower bound for the Woods invariant Nd > c d√log d was
proved. In the papers [6, 7] of U. Shapira and B. Weiss stable lattices were
suggested to replace well-rounded lattices in McMullen’s approach.
In this paper we construct counterexamples to the Woods conjecture only
using simpler lattices Dn. Then counterexamples to the Woods conjecture
in dimensions d ≥ 24 are constructed. In Regev–Shapora–Weiss’s approach
the 15 dimensional laminated lattice was used. From our construction the
Woods conjecture is unknown now for only 14 dimensions d = 10, 11, . . . , 23.
2. Counterexamples to the Woods conjecture in dimensions
d ≥ 24
As in [5] set C(L) = 4r2(L). The Woods conjecture can be re-formulated
as follows.
Conjecture 2.1 (The Woods conjecture). Let L be an unimodular lattice
then C(L) ≤ d.
Counterexamples to the Woods conjecture 725
The main idea of the construction of [5] is as follows. Let L = α1Λ15 +
α2Zn where Λ15 is the 15 dimensional laminated lattice, which is critical
and then well-rounded. They choose suitable α1 and α2 such that L is a
well-rounded unimodular lattice. Then
C(L) = α21C(Λ15) + α22C(Zn) = α21C(Λ15) + α22n
This gave counterexamples for n ≥ 15. Then counterexamples to the Woods
conjecture in dimensions d ≥ 30 were constructed. It is a surprise that
actually the root lattice Dm as follows gives us better counterexamples to
the Woods conjecture.
In the general case set L = α1L1 ⊕ α2Zn, where L1 is a dimension
m lattice with volume vol(L1) and the length of shortest non-zero lattice
vector λ(L1). If we want the lattice L unimodular, then αm1 αn2 = 1vol(L1) . If
α1λ1(L1) = α2, this lattice is well-rounded. Hence when
α1 =
( 1
vol(L1)λ(L1)n
) 1
m+n
and
α2 =
(
λ1(L1)m
vol(L1)
) 1
m+n
,
the lattice L is unimodular and well-rounded.
Set D′m = {(x1, . . . , xm) : x1 + · · · + xm = even}. It is well-known that
this is a well-rounded lattice and
λ1(D′m) =
√
2
The shortest lattice vectors in D′m is of the form (0, . . . ,±1, 0, . . . , 0,
±1, 0, . . . ). Its covering radius is r(D′m) =
√
m
2 and its volume is 2. We refer
to [1] for the detail. Set Dm = { 121/m (x1, . . . , xm) : x1 + · · ·+xm = even}. It
is clear that lattice vectors 121/m (0, . . . ,±1, 0, . . . , 0,±1, 0, . . . ) with length√
2
21/m are the shortest non-zero lattice vectors in Dm. They span R
m and
this is a well-rounded lattice in Rm. This is an unimodular lattice. Moreover
λ1(Dm) =
√
2
21/m , r(Dm) =
√
m
2
m+1
m
and C(Dm) = m22/m .
Let L = α1Dm + α2Zn,L. Then if α1 = 2−
n(m−2)
2m(m+n) and α2 = 2
m−2
2(m+n) , L
is a well-rounded unimodular lattice.
From α1 and α2 we have
C(L) = 2−
n+2
m+n ·m+ 2
m−2
m+n · n
When m=n+4, 2−
n+2
m+n = 2−
1
2 and 2
m−2
m+n = 2
1
2 . Then C(L) = 2− 12 (n+4)+
2
1
2n and C(L) > 2.12320343n+ 2
√
2. When n ≥ 10,
C(L) > 2n+ 4
726 Hao Chen, Liqing Xu
So when d = 2n + 4 ≥ 24, C(L) > d = 2n + 4. When m = n + 5 and
d = 2n+ 5 and a similar result can be proved when n ≥ 12. For dimensions
d = 25 and d = 27, a simple computation as follows leads to the conclusion.
The case of d = 25. We set m = 15, n = 10. Then C(L) = 2− 1225 · 15 + 2 1325 ·
10 > 10.75466 + 14.339 > 25.
The case of d = 27. We set m = 15, n = 12. Then C(L) = 2− 1427 · 15 + 2 1327 ·
12 > 10.47132 + 16.754119 > 27.
So we have the following result.
Theorem 2.2. Set α1 = 2−
n(m−2)
2m(m+n) and α2 = 2
m−2
2(m+n) . Then L = α1Dm +
α2Zn is unimodular and well-rounded. Moreover
(1) C(L) > m+ n when m = n+ 4, n ≥ 10;
(2) C(L) > m+ n when m = n+ 5, n ≥ 12;
(3) C(L) > m+ n when m = 15, n = 10;
(4) C(L) > m+ n when m = 15, n = 12.
Then the Woods conjecture is not true for dimensions d ≥ 24.
References
[1] J. H. Conway & N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der
Mathematischen Wissenschaften, vol. 290, Springer, 1988.
[2] P. M. Gruber & C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland
Mathematical Library, vol. 37, North-Holland, 1987.
[3] L. Kathuria & M. Raka, “On conjectures of Minkowski and Woods for n = 9”, https:
//arxiv.org/abs/1410.5743, 2014.
[4] C. T. McMullen, “Minkowski’s conjecture, well-rounded lattices and topological dimen-
sions”, J. Am. Math. Soc. 18 (2005), no. 3, p. 711-734.
[5] O. Regev, U. Shapira & B. Weiss, “Counterexamples to a conjecture of Woods”, Duke
Math. J. 66 (2017), no. 13, p. 2443-2446.
[6] U. Shapira & B. Weiss, “A volume estimate for the set of stable lattices”, C. R. Math.
Acad. Sci. Paris 352 (2014), no. 11, p. 875-879.
[7] ———, “Stable lattices and the diagonal group”, J. Eur. Math. Soc. 18 (2016), no. 8, p. 1753-
1767.
[8] A. C. Woods, “Covering six-space with spheres”, J. Number Theory 4 (1972), p. 157-180.
Hao Chen
The College of Information Science and Technology/Cyber Security, Jinan University
Guangzhou, Guangdong Province 510632, China
E-mail: haochen@jnu.edu.cn
Liqing Xu
The College of Information Science and Technology/Cyber Security, Jinan University
Guangzhou, Guangdong Province 510632, China
E-mail: lqxu1@jnu.edu.cn


Counterexamples to the Woods Conjecture in dimensions $d \ge 24$

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On the spectrum of irrationality exponents of Mahler numbers

Kolyvagin’s result on the vanishing of \protect \Sha(E/K)[p^\infty ] and its consequences for anticyclotomic Iwasawa theory

On eigenvalues of the kernel \protect \frac{1}{2} +\protect \lfloor \protect \frac{1}{xy}\protect \rfloor - \protect \frac{1}{xy}

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Counterexamples to the Woods Conjecture in dimensions $d \ge 24$

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