A uniform stability principle for dual lattices

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by  Martin Vodička

A UNIFORM STABILITY PRINCIPLE FOR DUAL LATTICES
7 MARTINVODICˇKAANDPAVOLZLATOSˇ
1
0
2 Abstract. Weproveahighlyuniformstabilityor“almost-near”theoremfor
dual lattices of lattices L ⊆ Rn. More precisely, we show that, for a vector
n x from the linear span of a lattice L ⊆Rn, subject to λ1(L) ≥λ> 0, to be
a ε-close to some vector from the dual lattice L⋆ of L, it is enough that the
J
inner products ux are δ-close (with δ <1/3) to some integers for all vectors

0 u ∈ L satisfying kuk ≤ r, where r > 0 depends on n, λ, δ and ε, only.
1 Thisgeneralizesananalogousresultprovedforintegralvectorlatticesin[13].
Theproofisnonconstructive,usingtheultraproductconstructionandaslight

] portionofnonstandardanalysis.
T
N
.
h Informally, a property of objects of certain kind is “stable” if objects “almost sat-
t isfying” this property are already “close” to objects having the property. Making
a
m precise the vague notions “almost satisfying” and “close” various rigorous notions
of stability can be obtained. The study of stability of functional equations origi-
[
natesfromaquestionaboutthe stabilityofadditivefunctions and, moregenerally,

1 ofhomomorphismsbetween metrizable topologicalgroups,askedby Ulam, cf. [16],
v
[20], [21]. Since that time it has developed to an established topic in mathemat-
8
ical and functional analysis and extended to a variety of (systems of) functional
4
5 equations—see, e.g., Rassias [17], Sz´ekelyhidi [19]. The study of stability of the
2 homomorphypropertywithrespectto the compact-opentopologywascommenced
0 bythe secondofthe presentauthors[22],[23]. Asystematicandgeneraltreatment
1. ofthe topic within “compactambience”making use ofnonstandardanalysisis due
0 to Anderson [1]. The recent survey article by Boualem and Brouzet [3] is worth
7 while to be consulted, too.
1
In the present paper we will prove the stability theorem for dual lattices of
:
v vector lattices announced in the abstract, as well as some closely related results.
i Intuitively, such a stability should mean that every vector x from the linear span
X
of L, behaving almost like a vector from the dual lattice L⋆ of L in the sense that
r
a all its inner products ux = u x +...+u x with vectors u from a “sufficiently
1 1 n n
big” subset of L are “sufficiently close” to some integer, is already “arbitrarily
close” to a vector y L⋆. Below is the natural precise formulation arising from
this account. In it s∈pan(L) denotes the linear subspace of Rn generated by L,
aZ = minc∈Z a c = min a a , a a denotes the distance of the real
| | | − | − ⌊ ⌋ ⌈ ⌉ −
number a from the set of all(cid:0)integers Z, and (cid:1)x = √xx is the usual euclidean
norm on Rn. k k
Theorem 0.1. Let L Rn be a vector lattice. Then, for each ε > 0, there exist
⊆
a δ >0 and an r >0 such that for every x span(L), satisfying ux δ for all
Z
∈ | | ≤
u L, u r, there is a y L⋆ such that x y ε.
∈ k k≤ ∈ k − k≤
2010 Mathematics Subject Classification. Primary11H06;Secondary11H31, 11H60,03H05.
Key words and phrases. Lattice,duallattice,stability,ultraproduct,nonstandardanalysis.
1
2 M.VODICˇKAANDP.ZLATOSˇ
Such a statement naturally raises the question how the parameters δ and r
depend on the parameters n and ε, given in advance, and some properties of the
lattice L. We, in fact, will prove a stronger and more uniform result, answering
partly this question. Namely, we will show that one can pick any δ (0,1/3);
∈
then r can be chosen depending on n, ε, δ and, additionally, the first Minkowski
minimum λ (L). The precise formulation is given in Theorem 5.2. On the other
1
hand, as the proof of this result uses the ultraproduct construction, it establishes
themereexistenceofsuchanr andgivesnoestimatehowbigitactuallyhastobe.
Theorem 5.2 generalizes an analogous result proved in [13] for integral lattices,
replacing the condition L Zn by introducing an additional parameter λ>0 and
⊆
requiring λ (L) λ. The mentioned result in [13] was obtained as a side product
1
≥
of a stability result for characters of countable abelian groups the proof of which
usedPontryagin-vanKampendualitybetweendiscreteandcompactgroupsandthe
utraproduct construction. Our present result is based on an intuitively appealing
almost-near Theorem 4.5 formulated in terms of nonstandard analysis which is
linkedtoitsstandardcounterpart(Theorem5.2)viathe ultraproductconstruction
appliedtoasequenceoflattices. Asaconsequence,Pontryagin-vanKampenduality
is eliminated from the proof. Additionally, the passage from stability of characters
tostabilityofduallatticesin[13]naturallyledtoaformulationintermsofthepair
of mutually dual norms x = x +...+ x and x = max(x ,..., x ).
k k1 | 1| | n| k k∞ | 1| | n|
Inour presentwork,startingrightawayfromlattices, the (equivalent)formulation
in terms of the (selfdual) euclidean norm x = x seems more natural.
k k k k2
1. Lattices and dual lattices
We assume some basic knowledge of lattices or, more generally, of “geometry of
numbers”. The readers can consult, e.g., Cassels [4], Gruber, Lekkerkerker [7] or
Lagarias [11], however, for their convenience we list here the definitions of most
notions we use and some facts we build on.
A subgroup L of the additive group Rn, where n 1, is called a lattice if it is
≥
discrete,i.e.,thereisaλ>0suchthat x y λforanydistinctvectorsx,y L.
Viewing Rn alternatively as a vector skpa−ce okr≥an affine space and its element∈s as
vectors or points, we speak of vector lattices or point lattices, respectively. The
dimensionofthelinearspacespan(X)generatedbyasetX Rn iscalledtherank
⊆
of X, i.e., rank(X) = dimspan(X). A full rank lattice is a lattice of rank equal
the dimension of the ambient space Rn. A body is a nonempty bounded connected
set C Rn which equals the closure of its interior. A body C is called centrally
⊆
symmetric if x C for any x C; it is called convex if ax+(1 a)y C for
− ∈ ∈ − ∈
any x,y C and a [0,1]. An example of a centrally symmetric convex body is
the euclid∈eanunit ba∈ll B = x Rn; x 1 . The Minkowski successive minima
{ ∈ k k≤ }
of L (with respect to the unit ball B) are defined by
λ (L)=inf λ R; λ>0, rank(L λB) k
k
{ ∈ ∩ ≥ }
for 1 k rank(L). In particular, λ (L) = inf x ; 0 = x L . The covering
1
≤ ≤ {k k 6 ∈ }
radius of L is defined by
µ(L)=inf r R; r >0, span(L) L+rB .
{ ∈ ⊆ }
In all these cases the infima are in fact minima.
A UNIFORM STABILITY PRINCIPLE FOR DUAL LATTICES 3
A basis of a vector lattice L Rn is an ordered m-tuple β = (v ,...,v ) of
1 m
⊆
linearly independent vectors from L which generate L as a group, i.e.,
L=grp(v ,...,v )= c v +...+c v ; c ,...,c Z .
1 m 1 1 m m 1 m
{ ∈ }
Obviously, in such a case rank(L) = m. In the proof of the fact that every lattice
has a basis the following elementary lemma, to which we will refer within short,
plays a key role.
Lemma1.1. Let L Rn bealatticeof rankm and(v ,...,v ), with k <m, bean
1 k
⊆
ordered k-tuple of linearly independent vectors from L which can be extended to a
basis of L. Denote V =span(v ,...,v ) and assume that the vector v LrV
1 k k+1
∈
has a minimal (euclidean) distance to the linear subspace V from among all the
vectors in LrV. Then the (k+1)-tuple (v ,...,v ,v ) either is already a basis
1 k k+1
of L (if k+1=m) or it can be extended to a basis of L (if k+1<m).
Another useful consequence of the fact that every lattice has a basis is the fol-
lowing
Lemma 1.2. Let L Rn be a lattice. Then a k-tuple of vectors v ,...,v L
1 k
is linearly independen⊆t if and only if, for any integers c ,...,c Z, the equa∈lity
1 k
∈
c v +...+c v =0 implies c =...=c =0.
1 1 k k 1 k
A basis (v ,...,v ) of a lattice L is Minkowski reduced if, for each k m, v is
1 m k
≤
the shortest vector from L such that the k-tuple (v ,...,v ) can be extended to a
1 k
basis of L. It is known that every lattice has a Minkowski reduced basis.
For any subset S Rn we denote by
⊆
AnnZ(S)= x Rn; u S: ux Z
{ ∈ ∀ ∈ ∈ }
the integral annihilator of S. Obviously, AnnZ(S) is a subgroup of Rn for every
S Rn, however, even if for a lattice L Rn, the integral annihilator AnnZ(L)
⊆ ⊆
need not be a lattice, unless rank(L) = n. The dual lattice of L (also called the
polar or reciprocal lattice) is defined as the intersection
L⋆ =AnnZ(L) span(L).
∩
Then L⋆ is a lattice in Rn of the same rank as L and there is an obvious duality
relation L⋆⋆ = L. The Minkowski successive minima of the original lattice L and
its dual lattice L⋆ are related through a bound due to Banaszczyk [2]. Similarly,
the covering radius of the dual lattice L⋆ can be estimated in terms of the first
Minkowski minimum of L—see Lagarias, Lenstra, Schnorr [12]. Actually, in the
quotedpaperstheseresultswerestatedandprovedforfullranklattices,i.e.,incase
m = n, only. However, introducing an orthonormal basis in the linear subspace
span(L)andreplacinganyvectorx span(L)byits coordinateswithrespecttoit,
∈
they can be readily generalized as follows.
Lemma 1.3. Let L Rn be a lattice of rank m. Then
⊆
λ (L)λ (L⋆) m
k m−k+1
≤
for each k m, and
≤
1
λ (L)µ(L⋆) m3/2.
1
≤ 2
4 M.VODICˇKAANDP.ZLATOSˇ
2. Ultraproducts of lattices
Inordertokeepourpresentationself-contained,wegiveabriefaccountoftheultra-
product construction and some notions of nonstandard analysis here. Nonetheless,
the readersarestronglyadvisedtoconsultsomemoredetailedexpositionlike,e.g.,
in Chang-Keisler [5], Davis [6] and Henson [8].
A nonempty system D of subsets of a set I is a called a filter on I if / D, D
∅∈
is closed with respect to intersections, and, for any X D, Y I, the inclusion
∈ ⊆
X Y implies Y D. A filter D on I is called an ultrafilter if for any X I
eith⊆er X D or Ir∈X D. Ultrafilters of the form D = X I; j X , wh⊆ere
∈ ∈ { ⊆ ∈ }
j I, are called trivial. As a consequence of the axiom of choice, every filter on I
∈
is contained in some ultrafilter; in particular, nontrivial ultrafilters exist on every
infinite set I.
Given a set I and a system of first order structures (A ) of some first order
i i∈I
language Λ, we can form their direct product A with basic operations and
i∈I i
relations defined componentwise. If, additionallQy, D is a filter on I, then
α β i I; α(i)=β(i) D
D
≡ ⇐⇒ { ∈ }∈
defines an equivalence relation on A . Denoting by α/D the coset of a function
i
α Ai with respect to D, theQquotient
∈ ≡
Q
B = A D = A ,
i i D
Y . Y .≡
naturally becomes a Λ-structure once we define
fB(α /D,...,α /D)=β/D,
1 p
where β(i)=fAi(α (i),...,α (i)), for any p-ary functional symbol f, and
1 p
(α /D,...,α /D) RB i I; (α (i),...,α (i)) RAi D
1 p 1 p
∈ ⇐⇒ ∈ ∈ ∈
(cid:8) (cid:9)
for any p-ary relational symbol R. Then B is called the filtered or reduced product
ofthe system(A ) with respectto the filter D. If A =A is the same structure for
i i
each i I, then the reduced product
∈
AI D = A D
i
(cid:14) Y .
is called the filtered or reduced power of the Λ-structure A. If D is an ultrafilter,
then we speak of ultraproducts and ultrapowers.
The key property of ultraproducts is the following
Lemma 2.1. [L os Theorem] Let (A ) be a system of structures of some first
i i∈I
order language Λ, D be an ultrafilter on the index set I, Φ(x ,...,x ) be a Λ-
1 p
formula and α ,...,α A . Then the statement Φ(α /D,...,α /D) holds in
1 p i 1 p
∈
the ultraproduct Ai D iQf and only if
Q (cid:14)
i I; Φ(α (i),...,α (i)) holds in A D.
1 p i
∈ ∈
(cid:8) (cid:9)
As a consequence, the canonic embedding of any Λ-structure A into its ultra-
power ∗A = AI D is elementary. More precisely, identifying every element a A
∈
with the coset a¯(cid:14)/D of the constant function a¯(i)=a, we have
Φ(a ,...,a ) holds in A Φ(a ,...,a ) holds in ∗A
1 p 1 p
⇐⇒
foreveryΛ-formulaΦ(x ,...,x ) andany a ,...,a A. This equivalence willbe
1 n 1 p
∈
referred to as the transfer principle.
A UNIFORM STABILITY PRINCIPLE FOR DUAL LATTICES 5
The above accounts almost directly apply to many-sorted structures, like mod-
ules overringsor vectorspacesoverfields, as well(see [8]). In particular,if (V)
i∈I
is a system of vector spaces overa field F, then the ultraproduct V D becomes
i
a vector space of the ultrapower ∗F =FI D, which is a field elemQenta(cid:14)rily extend-
ing F. Similarly, if (Gi)i∈I is a system of(cid:14)abelian groups, viewed as modules over
the ring of integers Z, then the ultraproduct G D becomes not only an abelian
i
group but also a module over the ring of hyQperin(cid:14)tegers ∗Z = ZI D, elementarily
extending the ring Z. And, what is of crucial importance, the L os(cid:14)Theorem is still
true for formulas in the corresponding two-sorted language.
Coming to the core,from now on I = 1,2,3,... denotes the set of all positive
{ }
integers and D is some fixed nontrivial ultrafilter on I. We form the ordered field
of hyperreal numbers as the ultrapower ∗R=RI D of the ordered field R. Then
(cid:14)
F∗R= x ∗R; r R, r >0: x <r ,
{ ∈ ∃ ∈ | | }
I∗R= x ∗R; r R, r >0: x <r
{ ∈ ∀ ∈ | | }
denote the sets of all finite hyperreals and of all infinitesimals, respectively. It can
beeasilyverifiedthatF∗Risasubringof ∗RandI∗RisanidealinF∗R. Hyperreal
numbers not belonging to F∗R are called infinite. For x ∗R we sometimes write
x < instead of x F∗R, and x instead of x / F∈∗R. Two hyperreals x, y
a|r|e sa∞id to be infinite∈simally close,∼in∞notation x y,∈if x y F∗R. Moreover,
foreachx F∗R, there is a unique realnumber ◦x≈ R, calle−dth∈e standard part of
x, such tha∈t x ◦x. As a consequence, F∗R/I∗R=∈R as ordered fields.
In terms of s≈equences, x=α/D, where α: I ∼R, is finite if and only if there is
a positive r R suchthat i I; α(i) <r →D; is it the case then the sequence
∈ { ∈ | | }∈
α converges to ◦x with respect to the filter D. In particular, x is infinitesimal if
and only if ◦x = 0, i.e., if and only if the sequence α converges to 0 with respect
to the filter D. As D necessarily extends the Frechet filter, lim α(i) = a R
i→∞
∈
in the usual sense implies ◦x=a, i.e., x a.
≈
The standard part map has the following homomorphy properties with respect
to the field operations:
◦(x+y)=◦x+◦y and ◦(xy)=◦x◦y
for any x,y F∗R, and if additionally x 0, then also ◦ x−1 = ◦x −1.
∈ 6≈
Alongwiththeequivalencerelationofinfinitesimalnea(cid:0)rness(cid:1) ,(cid:0)we(cid:1)introducethe
equivalence relation of order equality on ∗R as follows: ≈
∼
x
x 0 x=0, and x y 0 < for x,y =0.
∼ ⇔ ∼ ⇔ 6≈(cid:12)y(cid:12) ∞ 6
(cid:12) (cid:12)
(cid:12) (cid:12)
Thenx yispronouncedas“xandyareofthesam(cid:12) e(cid:12)order”. However,as / ∗R,
∼ ∞∈
one cannot infer x y from x and y . We also say that x is of smaller
∼ ∼ ∞ ∼ ∞
order than y or that y is of bigger order than x, in symbols x y, if y = 0 and
≪ 6
x 0. Obviously, for x,y =0, x y is equivalent to neither x y nor y x.
y ≈ 6 ∼ ≪ ≪
According to the transfer principle, we can identify the vector space (∗R)n over
the field ∗R and the ultrapower ∗(Rn) = (Rn)I D, so that the notation ∗Rn is
unambiguous. More generally, for any subset S (cid:14) Rn we identify the ultrapower
⊆
∗S =SI D with the subset
(cid:14)
(α /D,...,α /D) ∗Rn; i I; (α (i),...,α (i)) S D
1 n 1 n
∈ { ∈ ∈ }∈
(cid:8) (cid:9)
6 M.VODICˇKAANDP.ZLATOSˇ
of ∗Rn. In the same vein, the inner product Rn Rn R extends to the inner
product ∗Rn ∗Rn ∗R, preserving all its first×order→properties. However, in
order to disce×rn the l→inear spans with respect to the fields R and ∗R, respectively,
we denote the latter by
∗span(X)= a x +...+a x ; x ,...,x X, a ,...,a ∗R ,
1 1 n n 1 n 1 n
{ ∈ ∈ }
allowingX tobeanysubsetof ∗Rn,andcallittheinternallinear span of X. Anal-
ogously, we discern the lattice or subgroup, i.e., the Z-submodule grp(v ,...,v )
1 m
of Rn generated by vectors v ,...,v Rn, and the internal lattice internally
1 m
generated by vectors v ,...,v ∗Rn, i.∈e., the ∗Z-submodule
1 m
∈
∗grp(v ,...,v )= c v +...+c v ; c ,...,c ∗Z
1 m 1 1 m m 1 m
{ ∈ }
of ∗Rn.
Similarly as in ∗R, vectors from F∗Rn are called finite and vectors from I∗Rn
are called infinitesimal. Obviously,
F∗Rn = x ∗Rn; x < ,
∈ k k ∞
I∗Rn =(cid:8)x ∗Rn; x 0 (cid:9).
∈ k k≈
Both F∗Rn and I∗Rn are vector s(cid:8)paces over R and (cid:9)even modules over F∗R, but
not over ∗R. Vectors x,y ∗Rn are said to be infinitesimally close, in notation
x y, if x y I∗Rn,∈i.e., if x y 0. The standard part of a vector
x =≈ (x ,...,x− ) ∈F∗Rn is the vekcto−r ◦xk =≈ (◦x ,...,◦x ); obviously, ◦x is the
1 n 1 n
unique vector in R∈n infinitesimally close to x. Then F∗Rn/I∗Rn = Rn as vector
∼
spaces over R.
Though the ultraproduct construction can be applied to any system of lattices
L ∗Rni, it is sufficient for our purpose to deal with lattices situated in the same
i
am⊆bient space ∗Rn with n 1 fixed. Given a sequence (L ) of lattices L Rn
i i∈I i
≥ ⊆
we can form the ultraproduct L D and identify it with the subset
i
L= (α /D,...,α /D)Q∗R(cid:14)n; i I; (α (i),...,α (i)) L D
1 n 1 n i
∈ { ∈ ∈ }∈
of the vect(cid:8)or space ∗Rn over∗R. Then L is a “discrete” additive subgroupo(cid:9)f ∗Rn,
i.e., it is a module over the ring of hyperintegers ∗Z and it is discrete in the sense
that there is a positive λ ∗R such that x y λ for any distinct x,y L.
∈ k − k ≥ ∈
However, it should be noticed that λ may well be infinitesimal. Moreover, as D is
an ultrafilter, there is an m n and a set J D such that rank(L )=m for each
i
i J. We write rank(L) = m≤ and refer to L∈as an internal lattice in ∗Rn of rank
∈
m. Then we can assume, without loss of generality, that rank(L ) = m for each
i
i I. The sequence of Minkowski successive minima can be defined for such an
∈
internal lattice L in the natural way:
λ (L)= λ (L ) D
k k i i∈I
(cid:0) (cid:1) (cid:14)
for k m. Then 0 < λ (L) ... λ (L) is a sequence of hyperreal numbers,
1 m
≤ ≤ ≤
hence it cancontainboth infinitesimals as wellas infinite hyperreals. Additionally,
we put
rank (L)=# k; 1 k m, λ (L) 0 ,
0 k
{ ≤ ≤ ≈ }
rank (L)=# k; 1 k m, λ (L)< ,
f k
{ ≤ ≤ ∞}
where #H denotes the number of elements of a finite set H. We admit that
rank (L) = 0 if λ (L) 0, as well as rank (L) = 0 if λ (L) / F∗R. Obviously,
0 1 f 1
6≈ ∈
A UNIFORM STABILITY PRINCIPLE FOR DUAL LATTICES 7
if rank (L) > 0, then it is the biggest k m such that λ (L) 0; similarly, if
0 k
≤ ≈
rank (L)>0, then it is the biggest k m such that λ (L)< .
f k
≤ ∞
Atthesametime,wecanassumethatβ ,...,β L arefunctionssuchthat,
1 m i
∈
for each i I (or at least for each i from some set J QD), the m-tuple of vectors
∈ ∈
β(i)=(β (i),...,β (i)) is a Minkowskireduced basis of the lattice L . Then, due
1 m i
to L os Theorem (Lemma 2.1), the m-tuple β/D =(v ,...,v ), where v =β /D
1 m k k
for k m, is a Minkowski reduced basis of the internal lattice L, i.e., the vectors
v ,...≤,v are linearly independent over ∗R and generate L as a ∗Z-module.
1 m
Lemma 2.2. Let L ∗Rn be an internal lattice of rank m and β = (v ,...,v )
1 m
⊆
be a Minkowski reduced basis of L. Then the following hold true:
(a) If v v for some k <m and V =∗span v ,...,v , then
k k+1 1 k
xk kv≪k forkevery vector x LrV. { }
k+1
k k≪6 k k ∈
(b) v λ (L) for each k m.
k k
k k∼ ≤
Proof. (a) Assume that, under the assumptions of (a), we have x v for
k+1
some x LrV. For any vector y ∗Rn we denote y its orthkogkon≪alkprojecktion
V
to V. Le∈t us take a vectorz LrV∈such that its distance z z to V is minimal
V
from among all the vectors y∈ LrV. Therefore, −
∈
z z x x x ,
V V
k − k≤k − k≤k k
As z V, there are hyperreals a ,...,a ∗R such that z =a v +...+a v .
V 1 k V 1 1 k k
∈ ∈
Denoting c = a their lower integer parts and z′ = z c v ... c v L,
j j 1 1 k k
⌊ ⌋ − − − ∈
we have z z′ V, hence z′ z′ = z z , so that the vector z′ L has the
− ∈ k − Vk k − Vk ∈
same minimality property as z. Then, according to Lemma 1.1 and the transfer
principle, the (k+1)-tuple (v ,...,v ,z′) can be extended to a basis of L, hence
1 k
v z′ , as the basis (v ,...,v ) is Minkowski reduced. At the same time,
k+1 1 m
k k≤k k
z′ =(a c )v +...+(a c )v ,
V 1− 1 1 k− k k
with a c <1 for each j k. From the triangle inequality we get
j j
| − | ≤
z′ z′ + z′ z′
k k≤k Vk k − Vk
= (a c )v +...+(a c )v + z z
1 1 1 k k k V
k − − k k − k
< v +...+ v + x .
1 k
k k k k k k
Therefore, z′ v , and the more z′ < v , which is a contradiction.
k+1 k+1
k k≪k k k k k k
(b) As v = λ (L), the statement of (b) is true for k = 1. Assume, for a
1 1
k k
contradiction, that k <m for the biggest index satisfying v λ (L). Then
k k
k k∼
v λ (L)
k k+1
1 k k < and 0.
≤ λ (L) ∞ v ≈
k k+1
k k
Therefore,
v λ (L) v v λ (L)
k k+1 k k k+1
k k k k = k k 0.
v ≤ λ (L) · v λ (L) · v ≈
k+1 k k+1 k k+1
k k k k k k
Then, according to (a), kxk 0 for every vector x Lr∗span(v ,...,v ). In
kvk+1k 6≈ ∈ 1 k
particular, λk+1(L) 0. (cid:3)
kvk+1k 6≈
Remark 2.3. (b) of Lemma 2.2 follows immediately from the following estimates
of the lengths of vectors in any Minkowski reduced basis (v ,...,v ) of a rank m
1 m
8 M.VODICˇKAANDP.ZLATOSˇ
lattice L Rn in terms of its Minkowski successive minima, applying the transfer
⊆
principle:
λ (L) v 2kλ (L)
k k k
≤k k≤
for all k m (see Lagarias [10]; Mahler [14] has even better upper bounds). Then
≤
(a)couldbeprovedasaneasyconsequenceof(b). However,itisperhapsworthwhile
to notice that, using the internal lattice concept, the purely qualitative estimates
(a), (b) follow already from Lemma 1.1 and the existence of Minkowski reduced
bases.
The standard part or observable trace ◦X of a subset X ∗Rn consists of the
⊆
standardpartsofallfinitevectorsfromX;alternatively,itcanbeformedbytaking
justthefinitevectorsfromX and“mergingtogether”thosewhichareinfinitesimally
close, i.e., indiscernible. Identifying the results of both approaches, we have
◦X = X F∗Rn / = ◦x; x X F∗Rn = y Rn; x X: y x .
∩ ≈ ∈ ∩ ∈ ∃ ∈ ≈
In particu(cid:0)lar, for an(cid:1)additiv(cid:8)e subgroup G ∗R(cid:9)n we(cid:8)denote (cid:9)
⊆
FG=G F∗Rn and IG=G I∗Rn
∩ ∩
theadditivesubgroupsof ∗Rn formedbythefiniteandinfinitesimalelementsinG,
respectively. Then its standard part (observable trace) ◦G is an additive subgroup
of Rn which can be identified with the quotient
◦G=FG/IG.
However,evenforaninternallatticeL ∗Rn,itsstandardpart◦Lisnotnecessarily
discrete, hence it need not be a lattice⊆in Rn. A more detailed account will follow
after a preliminary lemma.
Lemma 2.4. Let L ∗Rn be an internal lattice of rank m and β = (v ,...,v )
1 m
⊆
be a Minkowski reduced basis of L such that all the vectors in β are infinitesimal.
Then there exist hyperintegers c ,...,c ∗Z such that all the vectors c v are
1 m k k
∈
finite but not infinitesimal and c divides c whenever 2 k m. For such
k k−1
≤ ≤
a choice of c , ..., c the internal sublattice M = ∗grp(c v ,...,c v ) L
1 m 1 1 m m
⊆
contains no infinitesimal vector except for 0, in other words λ (M) 0.
1
6≈
Proof. Let’s start with an arbitrary c ∗Z such that c v FLrIL (e.g., one
m m m
∈ ∈
can put c = v −1 guaranteeing that 1 c v <1+ v 1). Further
m m m m m
lk k m ≤k k k k≈
we proceed by backward recursion. Assuming that 2 k m and c is already
k
defined,weputc =c if c v 0(as v v≤ ,c≤v FLissatisfied
k−1 k k k−1 k−1 k k k−1
automatically),otherwise weput c 6≈ =bckwherke≤b k∗Zkis anyhy∈perintegersuch
k−1 k
∈
that bc v FLrIL (e.g., b = c v −1 will work). Obviously, c ∗Z
k k−1 k k−1 k
divides c ∈∗Z for any 2 k ml.k k m ∈
k−1
Assume th∈at x 0 where≤x=≤a c v +...+a c v for some a ,...,a ∗Z,
1 1 1 m m m 1 m
≈ ∈
not all equal to 0. Let q m be the biggest index such that a =0. Then
q
≤ 6
q
1 a c
x′ = x= k k v =0
k
c c 6
q kX=1 q
is a vector from the internal lattice L. Moreover, c x′ = x 0, while c v 0,
q q q
≈ 6≈
hence x′ v . Letp q bethesmallestindexsuchthat x′ v . Denote
q p
k k≪k k ≤ k k≪k k
λ = x′ if p = 1, or λ = max( v , x′ ) if p > 1. Then the hyperball λ∗B
p−1
k k k k k k
A UNIFORM STABILITY PRINCIPLE FOR DUAL LATTICES 9
contains p linearly independent vectors v ,...,v ,x′ from L, hence λ (L) λ
1 p−1 p
and, at the same time, λ v , contradicting Lemma 2.2(b). ≤(cid:3)
p
≪k k
Proposition 2.5. Let L= L D ∗Rn be an internal lattice of rank m and ◦L
i
⊆
be its observable trace. ThenQthe(cid:14)following hold true:
(a) ◦L is a lattice in Rn if and only if there is a positive λ R such that the set
∈
i I; λ (L ) λ belongs to D. This is equivalent to λ (L) 0 as well as
1 i 1
{ ∈ ≥ } 6≈
to rank (L)=0.
0
(b) ◦L is the direct sum of a linear subspace of Rn of dimension rank (L) and a
0
lattice in Rn of rank rank (L) rank (L).
f 0
−
(c) ◦L is a lattice of rank q m if and only if rank (L)=0 and rank (L)=q.
0 f
≤
Proof. (a) The equivalence of any of the first two conditions to the discreteness
of the group ◦L is obvious. Similarly, any of the obviously equivalent conditions
λ (L) 0 and rank (L) = 0 implies the discreteness of ◦L. Otherwise, there is
1 0
6≈
at least one nonzero infinitesimal vector v L. Then one can find a hyperinteger
c ∗Z such that cv is finite but not infin∈itesimal. Obviously, its standard part
w∈= ◦(cv) = 0 belongs to ◦L, so that span(w) = Rw is a line in Rn. We prove
the inclusio6n Rw ◦L. Taking any x = aw Rw, with a R, and putting
b = ac ∗Z, we⊆have b ac < b+1 which,∈by the virtue ∈of v 0, implies
⌊ ⌋ ∈ ≤ ≈
bv acv. Hence
≈ x=aw acv bv FL,
≈ ≈ ∈
and x = ◦(bv) ◦L. It follows that ◦L, containing the line Rw Rn, is not
∈ ⊆
discrete.
(b) Let (v ,...,v ) be a Minkowski reduced basis of L. Denote p = rank (L)
1 m 0
and q = rank (L). According to Lemma 2.2(b), a vector v is infinitesimal if and
f k
only if k p, and it is finite if and only if k q. For the same reason, if x
Lr∗grp(v≤,...,v ) then x v , hence x /≤FL. Therefore the standard par∈t
1 q q+1
k k≪6 ∈
◦LoftheinternallatticeLcoincideswiththestandardpartofitsinternalsublattice
∗grp(v ,...,v ). Due to Lemma 2.4, there are hyperintegers c ,...,c ∗Z such
1 q 1 p
that c v FLrIL for any k and c divides c for k 2. Then th∈e internal
k k k k−1
∈ ≥
sublattice M = ∗grp(c v ,...,c v ) L contains no nonzero infinitesimal vector.
1 1 p p
⊆
Letusdenotew =◦(c v )fork p,and,additionally,c =1,w =◦v =◦(c v )
k k k k k k k k
≤
for p<k q. As a consequence, ◦L coincides with the sum of the linear subspace
≤
span(w ,...,w ) and the lattice grp(w ,...,w ).
1 p p+1 q
The proof of (b) will be complete once we establish the following claim.
Claim. The vectors w ,...,w are linearly independent over R.
1 q
Indeed,letb ∗Nbeanyinfinitehypernaturalnumber. Putv′ =b−1v ,c′ =bc
∈ k k k k
foranyk q. Thenallthevectorsv′,...,v′ areinfinitesimalandformaMinkowski
≤ 1 q
reducedbasis ofthe lattice L′ = b−1x; x L . Now, allthe vectorsc′v′ =c v ,
∈ k k k k
where k q, are finite but not i(cid:8)nfinitesimal an(cid:9)d c′ divides c′ for k 2. From
≤ k k−1 ≥
Lemma 2.4 we infer that the internal lattice
N =∗grp(c v ,...,c v )=∗grp c′v′,...,c′v′
1 1 q q 1 1 q q
satisfies λ (N) 0. Then, by (a), its observa(cid:0)ble trace ◦N i(cid:1)s a lattice in Rn.
1
6≈
According to Lemma 1.2, it suffices to show that a w +...+a w = 0 implies
1 1 q q
a = ... = a = 0 for any integers a ,...,a Z. Since the first equality is
1 q 1 q
∈
equivalenttoa c v + +a c v 0andthelefthandvectorbelongstoN,which
1 1 1 q q q
··· ≈
contains no infinitesimal vector except for 0, we have a c v + +a c v = 0,
1 1 1 q q q
···
10 M.VODICˇKAANDP.ZLATOSˇ
and the desired conclusion follows from the linear independence of the vectors
c v ,...,c v over ∗R.
1 1 q q
(c) follows directly from (a) and (b). (cid:3)
Let us record the following direct consequence of (b).
Corollary 2.6. Let L be an internal lattice in ∗Rn. Then its observable trace ◦L
is a closed subgroup of the additive group Rn.
3. A linear-algebraic digression
Inthissectionweproveapreliminarynonstandardstabilityor“almost-near”result
concerningsolutionsofsystemsoflinearequations. Tothisendweneedtointroduce
some notions and notational conventions.
Given a field F we denote by Fm×n the vector space of all m n matrices over
×
F. Unless otherwise said, the vector space Fn consists of column vectors. Given a
vector x = (x ,...,x )T Fn and a subset J = j < ... < j 1,...,n we
1 n 1 q
∈ { } ⊆ { }
denotebyxJ Fq thevectorwithcomponentsxJ =x fori q. If n=q+sand
∈ i ji ≤
1,...,n = J K is a partition of the index set 1,...,n into two disjoint sets
{ } ∪ { }
J = j < ... < j , K = k < ... < k , then, for any vectors u Fq, v Fs,
1 q 1 s
{ } { } ∈ ∈
we denote by u v the unique vector x Fn satisfying xJ =u, xK =v.
J,K
⊔ ∈
It is a folklore (see, e.g., Strang [18]) that given a matrix A Fm×n and a
∈
vectorb Fm, the system oflinear equations Aξ =b can be solvedvia the Gauss-
∈
Jordan elimination, i.e., by applying a sequence of elementary row operations to
the augmented matrix (A b) of the system, transforming it into the reduced row
|
echelon form (B c), corresponding to the equivalent system Bξ = c out of which
|
allthe solutions canbe written downimmediately. However,the echeloncondition
is mainly of cosmetic nature and we can dispense with it, replacing it by a weaker
one. Let us call a T 1,...,m 1,...,n a transversal set of the matrix A if
⊆{ }×{ }
it satisfies the following four conditions:
(T1) if (i,j) T, then a =1 and a =0 for any k m, k =i;
ij kj
∈ ≤ 6
(T2) iftheithrowof Aisnonzero,thenthereisauniquej nsuchthat(i,j) T;
≤ ∈
(T3) if (i ,j ),(i ,j ) T and i <i , then j <j ;
1 1 2 2 1 2 1 2
∈
(T4) if (i,j) T and the kth row of A is identically zero, then i<k.
∈
A matrix A Fm×n is called transversally reduced if it has some (not necessarily
∈
unique) transversal set. Trivially, every matrix in reduced row echelon form is
transversally reduced.
Itisobviousthatapplyingaseriesofelementaryrowoperationstotheaugmented
matrix (A b) of the system Aξ = b, transforming it into a transversally reduced
|
matrix (B c), we obtain an equivalent system Bξ =c which has a solution if and
|
only if (B c) does not contain the row (0,...,0 1). Is it the case, then all the
| |
solutions can be described as follows: Let T be some transversal set of the matrix
B (hence of the matrix (B c), as well). Denote
|
I =dom(T)= i m; j n: (i,j) T ,
{ ≤ ∃ ≤ ∈ }
J =rng(T)= j n; i m: (i,j) T , and K = 1,...,n rJ.
{ ≤ ∃ ≤ ∈ } { }
Then the relation T is a bijection T: I J. Let further BI,K be the matrix
→
obtained from B by taking just its rows with indices belonging to I and columns
with indices belonging to K. Then a vector x Fn is a solution of the system
∈