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], Székelyhidi [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 0: x 0: x 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 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

A uniform stability principle for dual lattices

Similar A uniform stability principle for dual lattices