AA subspace theorem for manifolds
Emmanuel Breuillard and Nicolas de SaxcéFebruary 8, 2021
Abstract
We prove a theorem that generalizes Schmidt’s Subspace Theorem inthe context of metric diophantine approximation. To do so we reformulatethe Subspace theorem in the framework of homogeneous dynamics byintroducing and studying a slope formalism and the corresponding notionof semistability for diagonal flows.
Introduction
In 1972, Wolfgang Schmidt formulated his celebrated subspace theorem [32,Lemma 7], a far reaching generalization of results of Thue [37], Siegel [35], andRoth [28] on rational approximations to algebraic numbers. Around the sametime, in his work on arithmeticity of lattices in Lie groups, Gregory Margulis [26]used the geometry of numbers to establish the recurrence of unipotent flows onthe space of lattices GL d ( R ) / GL d ( Z ) . More than two decades later, a quantita-tive refinement of this fact, the so-called quantitative non-divergence estimate,was used by Kleinbock and Margulis [18] in their solution to the Sprindzukconjecture regarding the extremality of non-degenerate manifolds in metric dio-phantine approximation. As it turns out, these two remarkable results – thesubspace theorem and the Sprindzuk conjecture – are closely related and canbe understood together as statements about diagonal orbits in the space of lat-tices. In this paper we prove a theorem that generalizes both results at the sametime. We also provide several applications. This marriage is possible thanks toa better understanding of the geometry lying behind the subspace theorem,in particular the notion of Harder-Narasimhan filtration for one-parameter di-agonal actions, which leads both to a dynamical reformulation of the originalsubspace theorem and to a further geometric understanding of the family ofexceptional subspaces arising in Schmidt’s theorem. The proof blends the dio-phantine input of Schmidt’s original theorem with the dynamical input arisingfrom the Kleinbock-Margulis approach and refinements recently obtained in [2]and [8].We now formulate the main theorem. Let M = φ ( U ) be a connected analyticsubmanifold of GL d ( R ) parametrized by an analytic map φ : U → GL d ( R ) ,1 a r X i v : . [ m a t h . N T ] F e b here U ⊂ R n is a connected open set and n ∈ N . Let µ be the push-forward in M of the Lebesgue measure on U . The Zariski closure of M is said to be definedover Q if for ( a ij ) ij ∈ GL d the ideal of polynomial functions in C [ a ij , det − ] thatvanish on M can be generated by polynomials with coefficients in Q . Theorem 1 (Subspace theorem for manifolds) . Assume that the Zariski closureof M in GL d is defined over Q . Then there exists a finite family of propersubspaces V , . . . , V r of Q d with r = r ( d ) such that, for µ -almost every L in M ,for every ε > , the integer solutions x ∈ Z d to the inequality d (cid:89) i =1 | L i ( x ) | ≤ (cid:107) x (cid:107) − ε , (1) all lie in the union V ∪ · · · ∪ V r , except a finite number of them. Here the L i are the linear forms on R d given by the rows of L ∈ GL d ( R ) ,and (cid:107) · (cid:107) is the canonical Euclidean norm on R d .We will in fact prove the theorem under a slightly weaker assumption on M requiring only that what we call the Plücker closure of M be defined over Q ,see §1.8 for the definition of Plücker closure.Note that we recover the original Schmidt’s subspace theorem [33, 10, 5] asthe special case when M is a singleton { L } (then n = 0 and µ is the Diracmass at L ). On the other hand, even in the case where M is defined over Q ,the theorem is non-trivial. Indeed, as we shall see in Section 3.1, it recovers themain result of Kleinbock and Margulis regarding the Sprindzuk conjecture.The exceptional subspaces V i are independent of ε , as in Vojta’s refinement[38, 34], and they depend only the rational Zariski closure of M . In fact theyare determined by what we call the rational Schubert closure of M , that isthe intersection of all rational translates S σ g := BσB containing M , where g ∈ GL d ( Q ) and S σ is a standard Schubert variety associated to a permutation σ and a Borel subgroup B containing the diagonal subgroup. Each V i containsinfinitely many solutions to (1) , regardless of ε . The number r of exceptionalsubspaces can be bounded by a number depending only on d (see Lemma 3 andthe remark following it).The proof of Theorem 1 goes via the proof of a stronger result, a parametricsubspace theorem for manifolds, Theorem 3 below. This reformulates the prob-lem in terms of the dynamics of a one-parameter diagonal flow ( a t ) t> on thespace of lattices. We may summarize it informally as follows: Theorem 2 (Parametric version) . For µ -almost every L in M the lattice a t L Z d assumes a fixed asymptotic shape as t tends to + ∞ . By “fixed asymptotic shape” we mean two things. Firstly that the successiveminima are asymptotic to e Λ k t for some real numbers Λ k , Lyapunov exponentsof sorts, depending only on M and a = ( a t ) t> (the dependence on a is piecewiselinear). In particular as t varies there can only be oscillations of subexponentialsize for successive minima. And secondly that the successive minima determine2 fixed partial flag in Z d . In other words there is a fixed partial rational flag W ≤ . . . ≤ W d in Q d , such that if Λ k < Λ k +1 , then the k first successiveminima of a t L Z d are always realized by vectors from W k when t is large enough.The Λ k ’s and the flag { W k } k depend only on a and on the rational Schubertclosure of M . Grouping together the different W i obtained by varying theone-parameter subgroup a we obtain the family of exceptional subspaces V i appearing in Theorem 1.This rational flag arises naturally as the Harder-Narasimhan filtration as-sociated to a certain submodular function on the rational grassmannian: themaximal expansion rate of the subspace under the flow. We recall in §1.3 thatany submodular function on a grassmannian gives rise to a Grayson polygon, anotion of semistability, a Harder-Narasimhan filtration and certain coefficients,the slopes of the polygon, which in our case will correspond to the Lyapunovexponents Λ k mentioned above. This is the so-called “slope formalism”, whicharises in particular in the study of Euclidean lattices as first described by Stuhler[36] and Grayson [14], and in many other subjects as well [6, 27].Although we have restricted to the current setting for clarity of expositionin this introduction, the result will be proved for more general measures µ than push-forwards of the Lebesgue measure by analytic maps; the exceptionalsubspaces then depend only on the Zariski closure of the support of µ . Theright technical framework is that of good measures , which are closely related tothe friendly measures of [17], see §1.9.The paper is organized as follows. In Section 1 we begin by formulating thetechnical version of Theorem 2 and then proceed to describe the slope formalismon the grassmannian associated to a one-parameter flow and in particular discussthe associated notion of Harder-Narasimhan filtration. The proof of Theorems1 and 2 is carried out in §1.5 and §1.6 after a discussion of the Kleinbock-Margulis quantitative non-divergence estimates. In Section 2 we formulate andsketch a proof of an extension of Theorem 1 to arbitrary number fields, whichis analogous to the classical extension of Schmidt’s subspace theorem due toSchlickewei to multiple places and targets [31, 5, 39]. Finally in Section 3 weprove several applications of the main result.For the sake of brevity we do not state these applications in the introduc-tion and refer the reader to Section 3 directly instead. Let us only brieflymention that there are five main applications: ( i ) we explain how to recover theSprindzuk conjecture (Kleinbock-Margulis theorem) from Theorem 1, ( ii ) weestablish a manifold version of the classical Ridout theorem regarding approxi-mation by rationals whose denominators have prescribed prime factors, ( iii ) werecover the main results of [2] regarding (weighted) diophantine approximationon submanifolds of matrices showing that they hold also for submanifolds de-fined over Q (and not only over Q ), ( iv ) we prove an optimal criterion for strongextremality (Corollary 3), which answers in this case a question from [19, 4], ( v ) we prove a Roth-type theorem for non-commutative diophantine approxi-mation on nilpotent Lie groups, extending to algebraic points what was donefor Lebesgue almost every point in our previous work with Aka and Rosenzweig31, 2].Further applications and an extension of some of the results of this paperto other reductive groups and homogenous varieties can be found in the secondauthor’s forthcoming work [30]. A lattice, that is a discrete subgroup ∆ of rank d in R d , can be written: ∆ = Z u ⊕ · · · ⊕ Z u d , where ( u i ) ≤ i ≤ d is a basis of R d . And the space Ω of lattices can be identifiedwith the homogeneous space Ω (cid:39) GL d ( R ) / GL d ( Z ) . The position of a lattice ∆ in the space Ω , up to a bounded error, is describedby its successive minima λ (∆) ≤ · · · ≤ λ d (∆) , defined by λ i (∆) = inf { λ > | rk(∆ ∩ B (0 , λ )) ≥ i } , where rk( A ) for A ⊂ ∆ denotes the rank of the free abelian subgroup of ∆ generated by A , and B (0 , λ ) is the Euclidean ball of radius λ centered at theorigin in R d .Theorem 1 will be deduced from the following description of the asymptoticbehavior of the successive minima along a diagonal orbit of the lattice L Z d ,where L is a µ -generic point of M . Here, as in Theorem 1, M = φ ( U ) isthe image of a connected open set in some Euclidean space U ⊂ R n underan analytic map φ : U → GL d ( R ) , and µ is the push-forward under φ of theLebesgue measure on U . Theorem 3 (Strong parametric subspace theorem for manifolds) . Assume thatthe Zariski closure of M is defined over Q . Let ( a t ) t ≥ be a diagonal one-parameter semigroup in GL d ( R ) . Then there exist real numbers Λ ≤ Λ ≤· · · ≤ Λ d such that for µ -almost every L ∈ M and each k ∈ { , . . . , d } , lim t → + ∞ t log λ k ( a t L Z d ) = Λ k . (2) Moreover, if d < d < · · · < d h = d in { , . . . , d } are chosen so that Λ = · · · = Λ d < Λ d +1 = · · · = Λ d < · · · < Λ d h − +1 = · · · = Λ d , then there exist rational subspaces V (cid:96) , (cid:96) = 0 , . . . , h in Q d such that • dim V (cid:96) = d (cid:96) and { } = V < V < · · · < V h = Q d , for µ -almost every L ∈ M the first d (cid:96) successive minima of a t L Z d areattained in V (cid:96) provided t is large enough.In other words: for all ε > , for µ -almost every L ∈ M , there is t L,ε > suchthat for t > t L,ε , (cid:96) = 1 , . . . , h, and x ∈ Z d , (cid:107) a t L x (cid:107) ≤ e t (Λ d(cid:96) − ε ) ⇒ x ∈ V (cid:96) − . (3)When M is reduced to a singleton, the above theorem is a refinement of theparametric subspace theorem, often attributed to Faltings and Wüstholz [13,Theorem 9.1]. As shown below, it can also be obtained directly from Schmidt’ssubspace theorem in its original form. The two results are really equivalent.An important point to make is that (2) is a limit and not only a liminfor a limsup. This can be understood as saying that the diagonal orbit of alattice defined over Q has an asymptotic shape at infinity. Of course the λ k can fluctuate, but only up to a small exponential error. This is of course insharp contrast with what happens for certain specific values of L . Indeed it ispossible to construct matrices L for which the successive minima have an almostarbitrary behavior along a given diagonal orbit, see [29, Theorem 1.3] and [7,Theorem 2.2]. Corollary 1 (Parametric subspace theorem for manifolds) . Keep the same as-sumptions as in Theorem 3. Let a = ( a t ) t ≥ be a one-parameter diagonal semi-group in SL d ( R ) . There exists a proper subspace V ( a ) of Q d such that given ε > , for µ -almost every L ∈ M , there is t L,ε > such that if t > t L,ε and x ∈ Z d , (cid:107) a t L x (cid:107) ≤ e − εt ⇒ x ∈ V ( a ) . Proof.
Here we have assumed that a t is unimodular. By Minkowski’s theoremthe product of all d successive minima of a t L Z d is bounded above and belowindependently of t . In view of (2) , this implies that (cid:80) di =1 Λ i = 0 . So Λ d ≥ .Hence we can take V ( a ) = V h − in the notation of Theorem 3.The rational subspaces V i appearing in Theorem 3 depend only on ( a t ) t ≥ and on the rational Zariski closure of M , namely the intersection of the closedalgebraic subsets of GL d defined over Q and containing M . This will be clearfrom the proof of Theorem 3 given below, where a more precise description of V ( a ) and the V i will be given. As we will see, the filtration { V i } i is the Harder-Narasimhan filtration associated to M and ( a t ) t ≥ , and the Λ i are the slopesof the Grayson polygon . The next few paragraphs contain preparations towardsthe proof of Theorem 3 given at the end of this section.
In this subsection M is an arbitrary subset of GL d ( R ) and a = ( a t ) t ≥ a one-parameter diagonal semigroup. We write a t = diag( e A t , . . . , e A d t ) for some real5umbers A , . . . , A d . For a non-zero subspace V ≤ R d we define its expansionrate with respect to a by τ ( V ) := lim t → + ∞ t log (cid:107) a t v (cid:107) (4)where v represents V in an exterior power ∧ k R d . This quantity takes values inthe finite set of eigenvalues of log a in exterior powers. More precisely: τ ( V ) = max { I ( a ); I ⊂ [1 , d ] , | I | = k, v I (cid:54) = 0 } (5)where I ( a ) = (cid:80) i ∈ I A i and v I is the coordinate of v in the basis e I = e i ∧ . . . ∧ e i k of ∧ k R d , where I = { i , . . . , i k } , k = dim V . By convention we will also set τ ( { } ) = 0 . We leave it to the reader to check that if A ≥ . . . ≥ A d , then τ ( V ) = I V ( a ) , with I V := { i ∈ [1 , d ] , V ∩ F i > V ∩ F i +1 } (6)where F i = (cid:104) e i , . . . , e d (cid:105) and ( e , . . . , e d ) is the canonical basis of R d .For a subset M ⊂ GL d ( R ) , we set: τ M ( V ) := max L ∈ M τ ( LV ) . (7)Similarly we see readily that τ M ( V ) = max { I ( a ); I ⊂ [1 , d ] , | I | = dim V, ( L v ) I (cid:54) = 0 for some L ∈ M } . (8)From this formula it is clear that for all subspaces Vτ M ( V ) = τ Zar ( M ) ( V ) where Zar ( M ) is the Zariski closure. Lemma 1 (Submodularity of expansion rate) . Let M ⊂ GL d ( R ) and assumethat its Zariski closure is irreducible. Then the map V (cid:55)→ τ M ( V ) is submodularon the grassmannian, i.e. satisfies, for every pair of subspaces W , W , τ M ( W ) + τ M ( W ) ≥ τ M ( W ∩ W ) + τ M ( W + W ) . (9) Proof.
Given a subspace W in R d , it is clear from (8) that the Zariski closure of { L ∈ M, τ L ( W ) < τ M ( W ) } is a proper subset of Zar ( M ) . By irreducibility, wemay choose L in M such that τ M = τ L on all four subspaces W , W , W ∩ W and W + W . It is therefore enough to prove the lemma in the case M = { L } .Now let u be a vector representing U = W ∩ W in some exterior power of R d .Let also w (cid:48) and w (cid:48) be such that u ∧ w (cid:48) and u ∧ w (cid:48) represent W and W ,respectively. The subspace W + W is then represented by u ∧ w (cid:48) ∧ w (cid:48) , andmoreover, for every t , (cid:107) a t L ( u ∧ w (cid:48) ) (cid:107)(cid:107) a t L ( u ∧ w (cid:48) ) (cid:107) ≥ (cid:107) a t L u (cid:107)(cid:107) a t L ( u ∧ w (cid:48) ∧ w (cid:48) ) (cid:107) . Together with formula (4), this shows that τ L is submodular. Note that this iscompatible with the convention τ M ( { } ) = 0 .6 .3 Harder-Narasimhan filtration Submodular functions on partially ordered sets give rise to a “slope formalism”as in [36, 14, 13, 6, 27]. This is well known. In this paragraph we recall the mainfacts we need and for the reader’s convenience we give short proofs. The key tothem is the following submodularity lemma, which in implicit form goes back atleast to Stuhler [36] and Grayson [14] in the context of Euclidean sublattices andtheir covolume and which we rediscovered in [2] in the present context (subspacesand their expansion rate). Let k be a field and Grass( k d ) the Grassmannian ofnon-zero subspaces of k d . Let φ : Grass( k d ) → R be a submodular function,that is satisfying (9) with φ in place of τ M . Lemma 2 (Submodularity lemma) . There is a subspace V φ ∈ Grass( k d ) achiev-ing the infimum of φ ( V ) − φ (0)dim V , V ∈ Grass( k d ) , and containing all other suchsubspaces.Proof. Let I be that infimum. Without loss of generality, up to changing φ into φ − φ (0) , we may assume that φ (0) = 0 . We begin by observing that φ is boundedbelow: if ( V n ) n is a sequence of distinct subspaces of maximal dimension with φ ( V n ) → −∞ , pick a fixed line L with L (cid:54)⊂ V n for infinitely many V n ; bysubmodularity inf n φ ( V n + L ) = −∞ , contradicting the maximality of dim V n .So I is finite. For k ≥ , we set I k to be the same infimum restricted to thosesubspaces V with dim V ≥ k . There is a maximal k such that I = I k . If k = d , then we can take V φ = k d and there is nothing to prove. Otherwise let ε > so that I k +1 > I + 2 ε . If a subspace W satisfies φ ( W ) ≤ ( I + ε ) dim W, ( ∗ )then dim W ≤ k . By definition there is such a subspace with dim V φ ≥ k , callit V φ . If Z is another subspace with ( ∗ ), then φ ( Z + V φ ) ≤ φ ( Z ) + φ ( V φ ) − φ ( Z ∩ V φ ) ≤ ( I + ε )(dim Z + dim V φ ) − I dim( Z ∩ V φ ) ≤ I dim( Z + V φ ) + 2 ε dim( Z + V φ ) which forces dim( Z + V φ ) ≤ k , and hence Z ≤ V φ , as desired. Definition 1 (Semistability) . We say that k d is semistable with respect to φ if V φ = k d . Definition 2 (Grayson polygon) . Let P φ : [0 , d ] → R be the convex piecewiselinear function that is the supremum of all linear functions whose graph in [0 , d ] × R lies below all points (dim V, φ ( V )) , V ∈ Grass( k d ) ∪ { } . Its graph iscalled the Grayson polygon of φ .Let ( d i , f i ) , i = 0 , . . . , h be the vertices of the Grayson polygon with d = 0 and d h = d , that is the angular points, where the slope changes, i.e. for i =1 , . . . h − , s i < s i +1 , s i := f i − f i − d i − d i − . V V V V Figure 1: a Grayson polygonThe main result is the following:
Proposition 1 (Harder-Narasimhan filtration) . For each i = 0 , . . . , h , there isa unique k -subspace V i ≤ k d such that dim V i = d i and φ ( V i ) = f i . Moreoverthe subspaces V i are nested, i.e. V < V < · · · < V h = k d , forming theso-called Harder-Narasimhan filtration of φ . In particular, we see that k d is semistable if and only if its Harder-Narasimhanfiltration is the trivial one { } < k d . Remark.
Note that given any k -subspace V ≤ k d the function φ V ( W ) := φ ( W ) − φ ( V ) defined on the quotient k d /V , where W = W/V for any k -subspace W containing V is submodular on Grass( k d /V ) . It is clear from the propositionthat V i /V i − is semistable with respect to φ V i − for i = 1 , . . . , h , and that { V i /V } i ≥ is the Harder-Narasimhan filtration of V /V with respect to φ V . Proof.
The existence and uniqueness of V is exactly what the submodularitylemma tells us. Suppose V i has been defined. We may apply the submodularitylemma again to φ V i on the quotient k d /V i and thus obtain a subspace V i +1 containing V i strictly and such that the function ( φ ( V ) − φ ( V i )) / (dim V − dim V i ) reaches at V i +1 its unique minimum (cid:96) i +1 among subspaces V containing V i . Byconstruction (cid:96) i < (cid:96) i +1 for each i ≥ .We now need to show that the Grayson polygon coincides with the polygon P drawn out of the points (dim V i , φ ( V i )) . In other words we have to prove thatif V is a subspace of k d and i is such that dim V i ≤ dim V < dim V i +1 , then φ ( V ) − φ ( V i ) ≥ (cid:96) i +1 (dim V − dim V i ) . (10)If V i ≤ V , this is by definition of V i +1 . Otherwise V ∩ V i < V i and by inductionwe may assume that (dim( V ∩ V i ) , φ ( V ∩ V i )) lies above P . So φ ( V i ) − φ ( V ∩ V i ) ≤ (cid:96) i (dim V i − dim( V ∩ V i )) . (11)8oreover, again by definition of V i +1 we have φ ( V i + V ) − φ ( V i ) ≥ (cid:96) i +1 (dim( V i + V ) − dim V i ) . (12)On the other hand φ is submodular, so φ ( V i + V ) + φ ( V i ∩ V ) ≤ φ ( V i ) + φ ( V ) .Combining this with (11) and (12) we obtain: φ ( V ) ≥ φ ( V i ) + (cid:96) i +1 (dim( V ) − dim( V i ∩ V )) − (cid:96) i (dim V i − dim( V ∩ V i )) . (13)But (cid:96) i +1 ≥ (cid:96) i . So (10) follows.This shows the existence of the V i and the fact that they are nested. To seethe uniqueness note that if dim V = dim V i and φ ( V ) = φ ( V i ) but V (cid:54) = V i , then (cid:96) i ≥ (cid:96) i +1 in view of (13) , which is a contradiction.In the sequel, we apply this general theory by taking k = Q and φ ( V ) theexpansion rate τ M ( V ) defined in (7) on the grassmannian of rational subspaces.The above definition of semistability reads: Definition 3.
A non-zero rational subspace V in R d will be called M - semistable with respect to a = ( a t ) t ≥ if for every rational subspace W ≤ V , τ M ( W )dim W ≥ τ M ( V )dim V .
Similarly this yields the notions of Grayson polygon and Harder-Narasimhanfiltration of M with respect to a . Remark (unstable subspace) . In the case when det a = 1 , a subspace V with τ ( V ) < corresponds to a point v in some ∧ k R d , which is unstable with respectto a in the terminology of geometric invariant theory, i.e. its a -orbit contains in its closure. So R d is M -semistable if and only there are no unstable subspacesin the full grassmannian.Next we make a remark about the dependence of the Harder-Narasimhanfiltration with respect to the choice of one-parameter semigroup. It is easy tosee that the Grayson polygon depends continuously on a . This is not so forthe filtration, because new nodes can appear under small deformations, but thefollowing lemma shows that the subspaces involved remain confined to a fixedfinite family. Let b ( n ) be the ordered Bell number, that is the number of weakorderings (i.e. orderings with ties) on a set with n elements. Lemma 3.
Let M ⊂ GL d ( R ) with irreducible Zariski closure. There is a finiteset S M of rational subspaces of R d with | S M | ≤ b (2 d ) such that, as a = ( a t ) t ≥ varies among all one-parameter diagonal semigroups of GL d ( R ) , the subspaces V i ( a ) arising in the Harder-Narasimhan filtration all belong to S M .Proof. For I ⊂ [ d ] , let (cid:98) I ( a ) = | I | (cid:80) i ∈ I A i , where a t = diag( e A t , . . . , e A d t ) . Weclaim that the entire Harder-Narasimhan filtration of M depends on a only viathe ordering of the various (cid:98) I ( a ) for I ⊂ [ d ] . Namely if (cid:98) I ( a ) and (cid:98) I ( a (cid:48) ) define the9ame weak ordering on the family of subsets of [ d ] , then the filtrations coincide.To see the claim note that every slope ( τ M ( V ) − τ M ( W )) / (dim V − dim W ) for W ≤ V is equal to (cid:98) I ( a ) for some I (because I W ⊆ I V as follows from (6) and τ M can be replaced by τ L for some fixed L as in the proof of Lemma 1), andProposition 1 tells us that V i ( a ) is defined as the unique solution to an extremalproblem involving the comparison of slopes. So only their order matters. Sincethere are at most b (2 d ) possible orders, we are done.We also see from this proof that the slopes Λ i are continuous and piece-wise linear in log a and actually linear on each one of the cells cut out by thehyperplanes (cid:98) I ( a ) = (cid:98) I (cid:48) ( a ) for I, I (cid:48) ⊂ [1 , d ] . Remark.
The ordered Bell number b ( n ) grows super-exponentially with n .This gives a rather poor bound on the number of exceptional subspaces inTheorem 1, especially in view of Schmidt’s bound d d from [34]. A more refinedargument, which we do not include here and is based on the study of the setof permutations arising from the Schubert closure of M (see §1.8) allows toimprove this to (2 d ) d . We now describe the dynamical ingredient of the proof. Using the quantita-tive non-divergence estimates (see Theorem 5 below), Kleinbock showed in [16]the existence of a well-defined almost sure diophantine exponent for analyticmanifolds. As described in our previous work [2] with Menny Aka and LiorRosenzweig this holds for more general measures and the exponent actually de-pends only on the Zariski closure of the support of the measure, a propertycalled inheritance in this paper, because the measure inherits its exponent fromthe Zariski closure of its support. We will need the following version of theseresults:
Theorem 4 (Inheritance principle) . Let ( a t ) t ≥ be a one-parameter diagonalsemigroup in GL d ( R ) and for L ∈ GL d ( R ) , set µ k ( L ) := lim inf t → + ∞ (cid:88) j ≤ k t log λ j ( a t L Z d ) . Let U ⊂ R n a connected open set, φ : U → GL d ( R ) an analytic map, and µ theimage of the Lebesgue measure under φ . Let M := φ ( U ) and Zar ( M ) its Zariskiclosure in GL d ( R ) . Then for µ -almost every L in M and each k = 1 , . . . , d , µ k ( L ) = sup L (cid:48) ∈ Zar ( M ) µ k ( L (cid:48) ) . Proof.
A lemma of Mahler [25, Theorem 3], which is a simple consequence ofMinkowski’s second theorem in the geometry of numbers, asserts that λ ( ∧ k g Z d ) is comparable to (cid:81) j ≤ k λ j ( g Z d ) within multiplicative constants independent of g ∈ GL d ( R ) . Since the k -th wedge representation ρ k : GL( R d ) → GL( ∧ k R d ) is10n embedding of algebraic varieties, it maps Zar ( M ) onto Zar ( ρ k ( M )) . Theseobservations allow to reduce the proof to the case where k = 1 , which we nowassume. To this end we first recall the quantitative non-divergence estimates ina form established in [15]: Theorem 5. [15, Theorem 2.2] Let M = φ ( U ) and µ be as in Theorem 4.There are C, α > such that the following holds. Let ρ ∈ (0 , and t > , andlet B := B ( z, r ) be an open ball such that B ( z, n r ) is contained in U . Assumethat for each v , . . . , v k in Z d with w := v ∧ v ∧ · · · ∧ v k (cid:54) = 0 , sup y ∈ B (cid:107) a t φ ( y ) w (cid:107) > ρ k . Then for every ε ∈ (0 , ρ ] , we have: |{ x ∈ B ; λ ( a t φ ( x ) Z d ) ≤ ε }| ≤ C (cid:0) ερ (cid:1) α | B | . Given β ∈ R , we say that a subset S ⊂ GL d ( R ) satisfies the condition ( C β )if the following holds ∃ c > ∀ k ∈ { , . . . , d } , ∀ t > , ∀ w = v ∧ . . . ∧ v k ∈ ∧ k Z d \ { } , sup g ∈ S (cid:107) a t g w (cid:107) ≥ ce kβt . ( C β )And we set: β ( S ) := sup { β ∈ R | S satisfies ( C β ) } . (14)Note that by construction, if S ⊂ S (cid:48) , then β ( S ) ≤ β ( S (cid:48) ) .1st claim: For all L ∈ S , β ( S ) ≥ µ ( L ) . If S = φ ( B ) , where φ and B are as inTheorem 5, equality holds for µ -almost every L ∈ φ ( B ) . Proof of claim: If β > β ( S ) , then ( C β ) fails. This implies that there exists t arbitrarily large such that sup g ∈ S (cid:107) a t g w (cid:107) ≤ e kβt for some w (cid:54) = 0 . However byMinkowski’s theorem applied to the sublattice represented by a t g w , this meansthat sup g ∈ S λ ( a t g Z d ) ≤ e βt . Hence µ ( L ) ≤ β .The opposite inequality for S = φ ( B ) will follow from the quantitative non-divergence estimate combined with Borel-Cantelli. Let β < β ( φ ( B )) . Then ( C β )holds and, given δ > , Theorem 5 applies with ρ := ce βt and ε = e ( β − δ ) t sothat |{ x ∈ B ; λ ( a t φ ( x ) Z d ) ≤ e ( β − δ ) t }| ≤ Ce − αδt | B | . Summing this over all t ∈ N , we obtain by Borel-Cantelli that for almost ev-ery x ∈ B , λ ( a t φ ( x ) Z d ) ≥ e ( β − δ ) t if t is a large enough integer. But thisclearly implies that λ ( a t φ ( x ) Z d ) ≥ e ( β − δ/ t for all large enough t > . Hence µ ( φ ( x )) ≥ β − δ/ for Lebesgue almost every x ∈ B . Since δ > is arbitrary,this proves the first claim.Now we make the following key observation. For every bounded set S ⊂ GL d ( R ) and compact set K ⊃ S , β ( S ) = β ( Zar ( S ) ∩ K ) = β ( H ( S ) ∩ K ) . (15)11ere H ( S ) is the preimage in GL d ( R ) under ρ of the linear span H S of all ρ ( g ) , g ∈ S , where ρ is the linear representation with total space E = ⊕ dk =1 ∧ k R d .This follows immediately from the following claim:2nd claim: There is C = C ( S ) > such that for all w and t we have: sup g ∈H ( S ) ∩ K (cid:107) a t g w (cid:107) ≤ C sup g ∈ S (cid:107) a t g w (cid:107) . (16) Proof of claim:
We note that H S = H Zar ( S ) = H Zar ( S ) ∩ K , because S and Zar ( S ) ∩ K have the same Zariski closure. Now consider the space L ( H S , E ) of linear maps from H S to E . If X ⊂ H S is a bounded set that spans H S , thenthe quantity L (cid:55)→ sup A ∈ X (cid:107) L ( A ) (cid:107) defines a norm | · | X on L ( H S , E ) . Thereforefor any two such sets X, X (cid:48) there is a constant C X (cid:48) ,X > such that for all L ∈ L ( H S , ∧ k R d ) we have | L | X (cid:48) ≤ C X (cid:48) ,X | L | X . This applies in particular to the sets X := ρ ( S ) and X (cid:48) := ρ ( Zar ( S ) ∩ K ) . Now (16) follows by setting L : A (cid:55)→ a t A w , an element of L ( H S , E ) . This ends theproof of the second claim.We may now finish the proof of Theorem 4. Since M is connected, it followsfrom the first claim that the µ -almost sure value of µ ( L ) for L ∈ M is uniqueand well-defined and equals β ( φ ( B )) for any ball B as in Theorem 5. It isalso equal to sup L ∈ M µ ( L ) by the first part of the claim. However, since φ isanalytic, the Zariski closure of φ ( B ) coincides with Zar ( M ) . So (15) entails β ( φ ( B )) = β ( H ( M ) ∩ K ) for any compact K ⊃ φ ( B ) . But the first claimapplied to S = Zar ( M ) ∩ K implies that µ ( L ) ≤ β ( Zar ( M ) ∩ K ) for all L ∈ Zar ( M ) ∩ K . Since K is arbitrary, we get sup L ∈ Zar ( M ) µ ( L ) ≤ β ( Zar ( M ) ∩ K ) .The right-hand side is the µ -almost sure value of µ ( L ) , so this inequality is anequality. This ends the proof. Without loss of generality we may assume that A ≥ . . . ≥ A d , where A =diag( A , . . . , A d ) and a t = exp( tA ) . Let V < V < . . . < V h = Q d be the Harder-Narasimhan filtration associated to the submodular function τ M on the grassmannian of Q d . Let d (cid:96) = dim V (cid:96) and Λ k = τ M ( V (cid:96) ) − τ M ( V (cid:96) − ) d (cid:96) − d (cid:96) − if d (cid:96) − < k ≤ d (cid:96) . We need to show that for each (cid:96) = 1 , . . . , h and for µ -almostevery L the limit (2) holds when d (cid:96) − < k ≤ d (cid:96) and that for t large enough thefirst d (cid:96) − successive minima of a t L Z d are attained in V (cid:96) − . Suppose this hasbeen proved for all (cid:96) < i and let us prove it for (cid:96) = i .By Minkowski’s second theorem, for every L ∈ M lim sup t → + ∞ t (cid:88) k ≤ d i log λ k ( a t L Z d ) ≤ τ L ( V i ) ≤ τ M ( V i ) .
12n the other hand we already know that for µ -almost every L lim t → + ∞ t (cid:88) k ≤ d i − log λ k ( a t L Z d ) = (cid:88) k ≤ d i − Λ k = τ M ( V i − ) . (17)Hence lim sup t → + ∞ t (cid:88) d i −
Proof of Theorem 1.
Without loss of generality, we may assume that M is abounded subset of GL d ( R ) . In particular there is C = C ( M ) ≥ such that (cid:107) L x (cid:107) ≤ C (cid:107) x (cid:107) for every x ∈ R d and L ∈ M . We are going to show that thereis a finite set P , with | P | ≤ (2 Cd ε − ) d of one-parameter unimodular diagonalsemigroups a = ( a t ) t ≥ with the following property. If L ∈ M and x ∈ Z d is asolution to (1) such that the integer part t of ε d log (cid:107) x (cid:107) is at least , then thereis a ∈ P such that (cid:107) a t L x (cid:107) ≤ de − t . (26)To see this let (cid:96) i = log | L i ( x ) | . By (1) we have (cid:96) + . . . + (cid:96) d ≤ − dt . Notethat (cid:96) i ≤ log (cid:107) x (cid:107) + log C ≤ ( dε + log C ) t . Let (cid:96) (cid:48) i = max { (cid:96) i , − Dt } , where D = 5 d (log C + dε ) . If there is an index i such that (cid:96) i (cid:54) = (cid:96) (cid:48) i , then (cid:96) (cid:48) + . . . + (cid:96) (cid:48) d ≤− Dt + ( d − D/ d ) t ≤ − dt . We conclude that (cid:96) (cid:48) + . . . + (cid:96) (cid:48) d ≤ − dt always.Now define b i := d ( (cid:96) (cid:48) + . . . + (cid:96) (cid:48) d ) − (cid:96) (cid:48) i . Then b + . . . + b d = 0 , and for each i(cid:96) i + b i ≤ (cid:96) (cid:48) i + b i ≤ − t. On the other hand | (cid:96) (cid:48) i | ≤ Dt , so | b i | ≤ Dt . Let B be set of integers points in Z d with coordinates in [ − D, D ] . Choose ( n , . . . , n d ) ∈ B such that | b i t − n i | ≤ for all i . In particular | (cid:80) d n i | ≤ d/ . Changing some n i to the next or previousinteger if needed, we may ensure that (cid:80) d n i = 0 and | b i t − n i | ≤ . Then we set a t = diag( e n t , . . . , e n d t ) and let P be the finite set of such diagonal semigroups.Note that | P | ≤ (6 D ) d . Clearly (cid:96) i + n i t ≤ − t and (26) follows.Now we may apply Corollary 1 and conclude that for µ -almost every L ∈ M ,if x is a large enough solution of (1) , it must lie inside V ( a ) for some a in P .This shows that the number of exceptional rational subspaces is finite. Howeverby Lemma 3 above the subspace V ( a ) can take at most b (2 d ) possible values as a varies among all unimodular diagonal semigroups. This ends the proof. Remark.
Note that conversely each V ( a ) , and hence each V i in Theorem 1,contains infinitely many solutions to (1) for every ε > .15 emark. Furthermore the rational subspaces V ∪ · · · ∪ V r depend only on therational Zariski closure of M and not on the choice of L . And because theyare defined by a simple slope condition their height is effectively bounded interms of the height of Zar ( M ) . On the other hand, the finite set of exceptionalsolutions lying outside the V i depends on L and ε and there is no known boundon their height or number, see [12, Prop. 5.1]. When M is a single point it ishowever possible to group together the finitely many exceptional solutions intoanother set of proper subspaces whose number, but not height, can be effectivelybounded, see [11]. R What happens if we remove the assumption that the Zariski closure of M isdefined over Q in Theorem 3 ? Without this assumption, diagonal flow trajec-tories may not behave as nicely and typically no limit shape is to be expected.However we may give a simple upper and lower bound on the almost sure valueof µ k ( L ) for k = 1 , . . . , d , which exists by Theorem 4, in terms of the rationaland real Grayson polygons as we now discuss. So far we have only consideredthe rational Harder-Narasimhan filtration { V Q i } h Q i =0 and its rational polygon G Q with slopes s Q i , because we have restricted ourselves to considering the grass-mannian of rational subspaces. But we may also take k = R in §1.3 with thesame submodular function τ M . This yields a new Harder-Narasimham filtration { V R i } h R i =0 for the real field and a new Grayson polygon G R with slopes s R i , thatobviously lies below the rational polygon.Let as before M = φ ( U ) be the image of a connected open set U ⊂ R n under an analytic map φ : U → GL d ( R ) and µ the measure on M that is theimage of the Lebesgue measure on U . In this paragraph we no longer assumethat Zar ( M ) is defined over Q . Let µ k be the µ -almost sure value of µ k ( L ) as given by Theorem 4 and µ supk the supremum over all L ∈ M of µ supk ( L ) ,where µ supk ( L ) is defined by the same formula as µ k ( L ) with a limsup in place ofthe liminf. Consider the points ( k, µ k ) for k = 1 , . . . , d and interpolate linearlybetween them, so as to form a polygon G µ as in Fig 1. Similarly form G supµ with ( k, µ supk ) . Note that G supµ is convex (being a supremum of convex polygons), but G µ may not be. Proposition 2 (“Sandwich theorem”) . The polygons G µ , G supµ lie in between therational Grayson polygon G Q and the real Grayson polygon G R . In other words,for each k = 1 , . . . d , (cid:88) i ≤ k s R m i ≤ µ k ≤ µ supk ≤ (cid:88) i ≤ k s Q n i , where m i , n i defined by dim V R m i − ≤ i < dim V R m i and dim V Q n i − ≤ i < dim V Q n i .Proof. The upper bound follows from Minkowski’s theorem: the first d i succes-16 V Q V Q V Q V Q = V R V R V R V R V R Figure 2: G µ lies between the rational and the real polygonssive minima in a t L Z d are smaller than those attained in V Q i , so µ k ( L ) ≤ lim sup t → + ∞ t (cid:88) k ≤ d i log λ k ( a t L Z d ) ≤ τ L ( V Q i ) ≤ τ M ( V Q i ) . Since this holds for each i = 0 , . . . , h Q , and the polygons are convex, we get that G supµ lies below G Q .The lower bound follows from a modified version of the proof of Theorem 4that uses instead the refined quantitative non-divergence estimates for successiveminima already mentioned and established in [30, Chapter 6] or [24, Theorem5.3]. We now explain this briefly and refer to [30, Theorem 7.3.1] for the detailsof a more general statement. Let B be a small ball around some x ∈ U . Notethat φ ( B ) is Zariski dense in M and thus τ M = τ φ ( B ) . Let β k be the infimum ofall τ M ( W ) , where W ranges among real subspaces with dimension k . In viewof (8) this implies that there is I with I ( a ) ≥ β k such that ( L w ) I (cid:54) = 0 for some L ∈ M . By compactness of the grassmannian of k -dimensional subspaces thereis c > such that max I,I ( a ) ≥ β k sup L ∈ M (cid:107) ( L w ) I (cid:107) ≥ c (cid:107) w (cid:107) for all W . It followsin particular that sup L ∈ M (cid:107) a t L w (cid:107) (cid:29) e tβ k uniformly in t > and in non-zerointeger valued w . Now let γ k ≤ β k be the largest such function of k that isconvex in k . Then the exact same Borel-Cantelli argument used in the proof ofClaim 1 in Theorem 4, using instead the refined quantitative non-divergence inthe form of [24, Theorem 5.3] shows that for µ -almost every L in B , µ k ( L ) ≥ γ k .Since β d i = τ M ( V R i ) , we have shown that G µ lies above G R .This proposition also holds for measures µ on GL d ( R ) which are good in thesense of §1.9 below, see [30, Theorem 7.3.1].If Zar ( M ) is defined over Q , then by uniqueness of the Harder-Narasimhanfiltration, we see that the real and rational filtrations coincide. In particular,in this case all three polygons coincide. If Zar ( M ) is defined over Q , thenthe filtration over R is in fact defined over Q , and Theorem 3 asserts that G µ coincides with G Q . However G R may be different.17imilarly: Corollary 2.
Let M be as in Theorem 1, except we no longer assume that Zar ( M ) is defined over Q . Let a = ( a t = exp( tA )) t be a diagonal flow. Assumethat R d is M -semistable with respect to a (see Def. 3). Then for µ -almost every L ∈ M , lim inf t → + ∞ λ ( a t L Z d ) ≥ d Tr( A ) . In this paragraph we define the notion of Plücker closure of a subset M ⊂ GL d ( R ) and we explain why in Theorems 1 and 3 instead of assuming that Zar ( M ) is defined over Q , it is enough to assume that the Plücker closure of M is defined over Q .Let M ⊂ GL d ( R ) be a subset and ( E, ρ ) be the direct sum of the exteriorpower representations of GL d , namely E = ⊕ dk =1 ∧ k R d . We denote by H M the R -linear span in End( E ) of all ρ ( g ) , g ∈ M . We further define the Plückerclosure H ( M ) of M as the inverse image of H M under ρ in GL d ( R ) . Note that H ( M ) contains the Zariski closure Zar ( M ) of M . We say that H ( M ) is definedover Q if H N has a basis with coefficients in Q in the canonical basis of E . Anobvious sufficient condition for this to hold is to ask for Zar ( M ) to be definedover Q .We also say that M is Plücker irreducible if ρ ( M ) is not contained in a finiteunion of proper subspaces of H M in End( E ) . Clearly Zariski irreducibilityimplies Plücker irreducibility. It is clear that Plücker irreducibility of M isenough to guarantee that τ M is submodular by the argument of Lemma 1. It isalso clear that τ M = τ H ( M ) .For simplicity of exposition in this paper, we have chosen to state the as-sumptions in our main theorems in terms of the Zariski closure of M , but infact Theorems 1 and 3 hold assuming only that the Plücker closure is definedover Q . The proof is verbatim the same as the one we have given, except that inTheorem 4, we get the following slightly stronger statement: for µ -almost every L in M and each k , µ k ( L ) = sup L (cid:48) ∈H ( M ) µ k ( L (cid:48) ) . Again the proof of this equality is exactly the one given for Theorem 4 when k = 1 . However the reduction to the case k = 1 via Mahler’s lemma no longerworks here, because H ( M ) may differ from H ( ρ ( M )) . Instead one may usethe enhanced version of the quantitative non-divergence estimates already men-tioned in the proof of Proposition 2, namely [30, Chapter 6] or [24, Theorem5.3] in place of Theorem 5, which enables one to run the argument simultane-ously for all k . This shows that the almost sure value of each µ k ( L ) , and thusthe polygon G µ defined in the previous paragraph, depend only on the Plückerclosure of M .There is another notion of envelope of M that is also natural to consider,namely the intersection Sch ( M ) of all translates S σ g of Schubert varieties S σ = σB containing M . Here B is one of the d ! Borel subgroups containing thediagonal subgroup and the closure is the Zariski closure. This
Schubert closure contains the Plücker closure. It is easy to see from (6) that the submodularfunction τ M depends only on Sch ( M ) . Restricting to rational translates S σ g with g ∈ GL d ( Q ) one obtains the rational Schubert closure Sch Q ( M ) . Theasymptotic shape and the exceptional subspaces appearing in Theorems 1 and3 depend only on Sch Q ( M ) . A natural question we could not answer is whetheror not the main theorem remains valid under the (weaker) assumption that Sch ( M ) is defined over Q . The answer is clearly yes when Sch ( M ) is definedover Q as follows readily from Proposition 2. In this paragraph, we define a class of measures called good measures , which iswider than the family considered so far of push-forwards of the Lebesgue measureunder analytic maps, and for which Theorems 1, 3 and 4 continue to hold. Thisclass is very closely related to the so-called friendly measures of [17, 22]: insteadof being expressed in terms of an affine span, the non-degeneracy condition isdefined in terms of local Plücker closures.First we need to recall some piece of terminology. We fix a metric on GL d ( R ) ,say induced from the euclidean metric on the matrices M d ( R ) . Given two posi-tive parameters C and α , a real-valued function f on the support of µ is called ( C, α ) -good with respect to µ if for any ball B in GL d ( R ) and all ε > µ ( { x ∈ B, | f ( x ) | ≤ ε (cid:107) f (cid:107) µ,B } ) ≤ Cε α µ ( B ) , where (cid:107) f (cid:107) µ,B = sup x ∈ B ∩ Supp µ | f ( x ) | . The measure µ is doubling on a subset X ⊂ GL d ( R ) if there exists a constant C (cid:48) such that for every ball B ( x, r ) ⊂ X , µ ( B ( x, r )) ≤ C (cid:48) µ ( B ( x, r )) . Then we say that a Borel measure µ is locally good at L ∈ GL d ( R ) if there existsa ball B around L and positive constants C, α such that(i) The measure µ is doubling on B .(ii) For every k ∈ { , . . . , d } , for every pure k -vector w = v ∧· · ·∧ v k in ∧ k R d ,and every a ∈ GL d ( R ) , the map y (cid:55)→ (cid:107) ay · w (cid:107) is ( C, α ) -good on B withrespect to µ .Recall next that given a subset S ⊂ GL d ( R ) and a point x ∈ S , the localPlücker closure H x ( S ) of S at x is the intersection over r > of the Plückerclosures of S ∩ B ( x, r ) : H x ( S ) = (cid:92) r> H ( S ∩ B ( x, r )) . We may now define the class of good measures.19 efinition 4.
A locally finite Borel measure µ on GL d ( R ) will be called a good measure if Supp µ is Plücker irreducible and if it satisfies the followingassumptions for µ -almost every L :1. The measure µ is locally good at L ;2. The local Plücker closure of Supp µ at L is equal to that of Supp µ .Of course, the most important example of a good measure is that of thepush-forward of the Lebesgue measure under an analytic map. Proposition 3 (Analytic measures are good) . Let n ∈ N , let U be a connectedopen set in R n , and let ϕ : U → GL d ( R ) be a real-analytic map. Then thepush-forward under ϕ of the Lebesgue measure on U is a good measure on theZariski closure M of ϕ ( U ) .Proof. Note that the maps of the form u (cid:55)→ (cid:107) aϕ ( u ) · w (cid:107) are linear combinationsof products of matrix coefficients of ϕ ( u ) and hence belong to a finite dimensionallinear subspace of analytic functions on U independent of the choice of a ∈ GL d ( R ) and w ∈ ∧ ∗ R d . So [16, Proposition 2.1] applies.Theorem 4 holds for all good measures µ on GL d ( R ) with the same proof(suitably modified via the enhanced quantitative non-divergence estimates asmentioned in the previous paragraph). In fact the definition of good measureshas been tailored precisely for Theorem 4 to hold. Thus the subspace theoremfor manifolds and the parametric subspace theorem, Theorems 1 and 3, hold forgood measures µ such that the Plücker closure of Supp( µ ) is defined over Q . Schmidt himself observed in [32] that his theorem for the field Q of rationalsimplied a more general version for any number field K , and this was generalizedshortly after by Schlickewei [31], who gave a statement allowing also finite places.In this section, we formulate a similar generalization of Theorems 1 and 3.For a place v of a number field K , the completion of K at v is denoted by K v . As in Bombieri-Gubler [5, § 1.3-1.4] we use the following normalization forthe absolute value | · | v on K v | x | v := N K v / Q v ( x ) K : Q ] , where Q v is the completion of Q at the place v restricted to Q and N K v / Q v ( x ) is the norm of x in the extension K v / Q v . The product formula then reads (cid:81) v | x | v = 1 for all x ∈ K . If S is a finite set of places of K containing allarchimedean ones, the ring O K,S ⊂ K of S -integers is the set of x ∈ K suchthat | x | v ≤ for all places v lying outside S . Elements of its group of units arecalled S -units. 20et d be a positive integer. The d -dimensional space K dv will be endowedwith the supremum norm (cid:107) · (cid:107) v , i.e. for an element x = ( x ( v )1 , . . . , x ( v ) d ) in K dv , (cid:107) x (cid:107) v = max ≤ i ≤ d | x ( v ) i | v . More generally, we let K S = (cid:81) v ∈ S K v be the productof all completions of K at the places of S , and if x = ( x ( v ) ) v ∈ S is an element of K dS , we define its norm (cid:107) x (cid:107) , its content c ( x ) and height H ( x ) by (cid:107) x (cid:107) = max v ∈ S (cid:107) x ( v ) (cid:107) v , c ( x ) = (cid:89) v ∈ S (cid:107) x ( v ) (cid:107) v and H ( x ) = (cid:89) v ∈ S max { , (cid:107) x ( v ) (cid:107) v } . It is clear that c ( x ) ≤ H ( x ) and (cid:107) x (cid:107) ≤ H ( x ) ≤ max { , (cid:107) x (cid:107) | S | } . It followsfrom the product formula that (cid:107) x (cid:107) ≥ if (cid:54) = x ∈ O K,S . The image of O K,S in K S under the diagonal embedding is discrete and cocompact in K S [23, ChapterVII], and closed balls in K S for the norm are compact. Furthermore, it is easilyseen from Dirichlet’s unit theorem that there is a constant C = C ( K, S ) > such that for all x ∈ K dS with c ( x ) (cid:54) = 0 , there is an S -unit α ∈ O K,S such that e − C (cid:107) α x (cid:107) | S | ≤ c ( x ) ≤ (cid:107) α x (cid:107) | S | . (27)For each v ∈ S , we denote by GL d ( K v ) the group of invertible d × d matriceswith coefficients in K v , and we set GL d ( K S ) = (cid:89) v ∈ S GL d ( K v ) . A product measure µ = ⊗ v ∈ S µ v on GL d ( K S ) will be called a good measure if each µ v is a good measure on GL d ( K v ) . The definition of a good measuregiven in §1.9 for K v = R extends verbatim to other local fields K v .Examples of good measures are provided by push-forwards of Haar measureunder strictly analytic maps. Indeed Proposition 3 continues to hold for analyticmaps φ : U → GL d ( K v ) whose coordinates are defined by convergent powerseries on a ball U := B ( x, r ) = { y ∈ K nv , max ni =1 | y i − x i | ≤ r } in K nv . Thisis because, on the one hand the push-forward of Haar measure on K v under φ will be locally good everywhere by [22, Prop. 4.2], and on the other hand, thePlücker closure of the image of a ball B ⊂ U of positive radius is independentof the ball, because convergent power series that vanish on an open ball mustvanish everywhere.Let E v := ⊕ k ≤ d ∧ k K dv . We will say that the Plücker closure of µ is definedover Q if for each v ∈ S , the subspace H Supp µ v of End( E v ) (see §1.8) is definedover Q , i.e. is the zero set of a family of linear forms on End( E v ) with coefficientsin Q ∩ K v . Clearly this is the case if for each v ∈ S , the Zariski closure of Supp µ v in GL d ( K v ) is defined over Q . We will also denote by H ( µ ) the cartesian productof all H ( µ v ) , where H ( µ v ) := { g ∈ GL d ( K v ) , ρ v ( g ) ∈ H Supp µ v } .We are now ready to state: Theorem 6 (Subspace theorem for manifolds, S -arithmetic version) . Let K bea number field, S a finite set of places including all archimedean ones, O K,S itsring of S -integers, and d in N . Let µ be a good measure on GL d ( K S ) whose lücker closure is defined over Q . Then there are proper subspaces V , . . . , V r of K d such that for µ -almost every L and for every ε > , the inequality (cid:89) v ∈ S d (cid:89) i =1 | L ( v ) i ( x ) | v ≤ c ( x ) ε (28) has only finitely many solutions x ∈ O dK,S \ ( V ∪ · · · ∪ V r ) up to multiplicationby an S -unit. Here L = ( L ( v ) ) v ∈ S ∈ GL d ( K S ) and each L ( v ) i denotes the i -th row of thematrix L ( v ) ∈ GL d ( K v ) . Note that the left and right-hand sides of (28) areunchanged if x is changed into α x for some S -unit α , so we will focus on theequivalence classes of solutions. The bound on the number r of exceptionalsubspaces depends only on d and | S | , and each subspace contains infinitelymany (classes of) solutions to (28) .When µ is a Dirac mass at a point L = ( L ( v ) ) v ∈ S with L ( v ) ∈ GL d ( Q ) ∩ GL d ( K v ) , then the theorem is exactly the S -arithmetic Schmidt subspace the-orem as stated in [5, 7.2.5]. Theorem 6 is deduced from Theorem 7 below, which is a parametric versionanalogous to Theorem 3. To formulate it, we need to define the S -arithmeticanalogues of a lattice and its successive minima. A family of vectors x , . . . , x k in K dS is said to be linearly independent if it spans a free K S -submodule of rank k . Equivalently the vectors x ( v )1 , . . . , x ( v ) k are linearly independent over K v foreach place v . Definition 5 (Lattice in a number field) . For any positive integers k ≤ d , wedefine a sublattice in K dS of rank k to be a discrete free O K,S -submodule of rank k in K dS . In other words, it is a subgroup ∆ ≤ K dS that can be written ∆ = O K,S x ⊕ · · · ⊕ O K,S x k for some linearly independent elements x i = ( x ( v ) i ) v ∈ S in K dS . If k = d we saythat ∆ is a lattice in K dS .If L is any element of GL d ( K S ) , then ∆ = L O kK,S is a sublattice of rank k in K dS . Conversely they are all of this form. Note that if x , . . . , x k are vectorsfrom a lattice ∆ , they are linearly independent if and only if x ( v )1 , . . . , x ( v ) k arelinearly independent for some place v . Definition 6 (Successive minima) . If ∆ is a sublattice in K dS and k ∈ [1 , d ] , λ k (∆) = inf { λ > | ∃ x , . . . , x k ∈ ∆ linearly independent with ∀ j, c ( x j ) ≤ λ } is its k -th successive minimum , where c ( x j ) is the content defined earlier.22t follows from (27) that we may have defined the successive minima using (cid:107) x j (cid:107) | S | in place of c ( x j ) without much difference. In particular either definitionwill suit the theorem below.The analogue of Theorem 3 now reads as follows. Fix a diagonal element a = diag( a ( v ) i ) in GL d ( K S ) , and consider the flow a t = a t , for every t ∈ N . Theorem 7 ( S -arithmetic strong parametric subspace theorem) . Let K be anumber field and S a finite set of places containing all archimedean ones. Let ( a t ) t ∈ N be a diagonal flow in GL d ( K S ) and µ a good measure on GL d ( K S ) whosePlücker closure is defined over Q . Then there are K -subspaces V < V < · · · < V h = K d and real numbers s < . . . < s h such that for each i = 1 , . . . , h and for µ -almost every L ,1. If dim V i − < k ≤ dim V i , then lim t → + ∞ t log λ k ( a t L ( O K,S ) d ) = s i .2. For all t > large enough, the first dim V i successive minima of a t L ( O K,S ) d are attained in V i .In other words for every ε > and µ -almost every L , there is t ε,L such that if t > t ε,L , (cid:96) = 1 , . . . , h, and x ∈ ( O K,s ) d , c ( a t L x ) ≤ e t ( s (cid:96) − ε ) ⇒ x ∈ V (cid:96) − . (29)The K -subspaces V i , i = 1 , . . . , h , appearing in Theorem 7 are the terms ofthe Harder-Narasimhan filtration associated to M := Supp µ ⊂ GL d ( K S ) andthe quantities s i are its slopes. We describe this filtration in the next paragraph. K dS In this paragraph we associate a submodular function on the grassmannian
Grass( K d ) that generalizes the expansion rate τ M defined earlier in §1.2. Givena K S -submodule V in K dS we define its expansion rate as follows τ ( V ) := lim t → + ∞ t log c ( a t v ) where v ∈ ∧ ∗ K dS = (cid:81) v ∈ S ∧ ∗ K v and c ( v ) is the content as defined earlier. Herewe identify ∧ k K v with K Nv , N = (cid:0) dk (cid:1) and use the standard basis e i ∧ . . . ∧ e i k to define the norm, and note that V = (cid:81) v V v , where each V v is a K v -vectorsubspace represented by an element v v ∈ ∧ ∗ K v . So τ ( V ) is just the sum of theexpansion rates of V v in K dv over all places v ∈ S . For a subset M ⊂ GL d ( K S ) we also define, τ M ( V ) = max L ∈ M τ ( LV ) . As in (8) above, we see that if V represented by v ∈ ∧ ∗ K dS , τ M ( V ) = max { I ( a ); I = ( I v ) v ∈ S , ∃ L ∈ M, ∀ v ∈ S, ( L ( v ) v ( v ) ) I v (cid:54) = 0 } , I v ⊂ [1 , d ] , | I v | = dim V v , I ( a ) = (cid:80) v ∈ S I v ( a ) and I v ( a ) = (cid:80) i ∈ I v log | a ( v ) i | ,and w I is defined by the expression w = (cid:80) | I | = k w I e I , where e I = e i ∧ . . . ∧ e i k when I = { i , . . . , i k } .We will say that M is Plücker irreducible if its projection to GL d ( K v ) isPlücker irreducible for each v ∈ S . Under this assumption we see by the sameargument as in Lemma 1 that τ M is submodular on the set of all K S -submodulesof K dS .If we restrict τ M to the set of K -subspaces of K d , i.e. the grassmannian Grass( K d ) , then we thus obtain a well-defined notion of Harder-Narasimhanfiltration, Grayson polygon and slopes. This gives what we will call the rationalGrayson polygon G K . And a K -linear space V will be M -semistable for thesemigroup ( a t ) if for every K -subspace W < V , τ M ( W )dim W ≥ τ M ( V )dim V .
But we may also consider τ M as a function on the set of all K S -submodulesof K dS . Since the dimension of each projection to K v may not be the same for all v , we use the following definition for the dimension of a submodule V = (cid:81) v V v : dim V = 1 | S | (cid:88) v ∈ S dim K v V v Then dim V is a modular function on the “full grassmannian”, i.e. the set of all K S -submodules of K dS . Thus Proposition 1 and its proof are still valid and weobtain a “full” Harder-Narasimhan filtration and a “full” Grayson polygon G K S ,whose nodes now have x -coordinates in | S | N . In this section we discuss the proof of Theorems 6 and 7. A basic ingredient isthe S -arithmetic version of Minkowski’s second theorem, which without payingattention to numerical constants, takes the following form: Theorem 8 (Minkowski’s second theorem) . Let ∆ be a sublattice in K dS as in (5) . Then c ( x ∧ . . . ∧ x k ) K : Q ] ≤ λ (∆) · . . . · λ k (∆) (cid:28) c ( x ∧ . . . ∧ x k ) K : Q ] , where the constant involved in the Vinogradov notation (cid:28) depends only on K , S and d , not on ∆ . The content c ( x ∧ . . . ∧ x k ) is proportional to the covolume of ∆ in its K S -span. See [5, Theorem C.2.11, page 611] for a proof when S has no non-archimedean places and [21] in the general case with the caveat that the nor-malizations used in the latter paper differ from ours especially at the complexplace, leading to a slightly different definition of the successive minima.24he derivation of Theorem 6 from Theorem 7 works verbatim as that ofTheorem 1 from Theorem 3 given in §1.6. One replaces d with d | S | and treatsall linear forms L ( v ) i on an equal footing. The constant D needs to be increasedappropriately and at non-archimedean places v the exponential used in thedefinition of the flow will be replaced by a power of a uniformizer π v of K v ,namely a ( v ) t = diag( π n ,v tv , . . . , π n d,v tv ) for integers n i,v . One needs also to recallthat in view of (27) , for each T > there are only finitely many classes of x ∈ O dK,S with c ( x ) ≤ T , so we may assume that c ( x ) is large. The number r of distinct V i obtained is similarly bounded by b (2 d | S | ) as in Lemma 3.Now the proof of Theorem 7 is again verbatim as that of Theorem 3, treatingall L ( v ) i on an equal footing and keeping the argument unchanged. One needsto invoke the S -arithmetic subspace theorem (in the form of Theorem 6 for asingle point, or as [5, 7.2.5]) in place of the ordinary subspace theorem. For thedynamical ingredient at the end, one applies instead the following generalizationof Theorem 4. Theorem 9 (Inheritance principle) . Let K be a number field, S a finite set ofplaces containing all archimedean places, and d in N . Let ( a t ) be a diagonalone-parameter semigroup in GL d ( K S ) , and µ a good measure on GL d ( K S ) with H ( µ ) = (cid:81) v ∈ S H ( µ v ) the Plücker closure of its support. For k ∈ [1 , d ] let µ k ( L ) := lim inf t →∞ t (cid:88) i ≤ k log λ i ( a t L O dK,S ) for each L ∈ GL d ( K S ) . Then, for µ -almost every L , µ k ( L ) = sup L (cid:48) ∈H ( µ ) µ k ( L (cid:48) ) . Sketch of proof.
When k = 1 the proof of this result is identical to the proof wegave of Theorem 4 with the following adjustments. Theorem 5 was extendedto this context by Kleinbock-Tomanov in [22, §8.4] (technically speaking onlyfor K = Q , but the general case is entirely analogous). The norm there and inthe definition ( C β ) of β must be replaced by the content of the correspondingvector. Theorem 8 must be used in place of the original Minkowski theorem toprove the first claim. Also (16) continues to hold with the content in place of thenorm for subsets of GL d ( K S ) that are cartesian products of subsets of GL d ( K v ) for v ∈ S , which is the case for Supp( µ ) by definition of a good measure. Sothe second claim also holds. The case k > is analogous, but as in the proofof Prop. 2, one needs to use the refined quantitative non-divergence estimateproved in [30, Chapter 6] and [24, Theorem 5.3] instead of Theorem 5.We end this section by stating the analogue of Proposition 2 in the S -arithmetic context. The following describes what is left of Theorem 7 if weremove the assumption that the Plücker closure of the good measure µ is de-fined over Q . In §2.2 we have defined two Grayson polygons: the rational one25 K coming from Grass( K d ) and the full one G K S coming from the full grass-mannian of all K S -submodules. And of course we have as before the polygon G µ with nodes ( k, µ k ) , k ∈ [1 , d ] , where µ k is the µ -almost sure value of µ k ( L ) provided by Theorem 9 and the polygon G supµ with nodes ( k, µ supk ) , where µ supk is the supremum of µ supk ( L ) over L ∈ M and µ supk ( L ) is defined as µ k ( L ) witha limsup in place of liminf. Proposition 4 ( S -arithmetic sandwich theorem) . The polygons G µ , G supµ lie inbetween G K and G K S , i.e. G K S ≤ G µ ≤ G supµ ≤ G K . Again the proof is mutatis mutandis that of Proposition 2.
In this last section we present a number of applications of the main theorem.
We begin by the demonstration of how the main result of Kleinbock and Mar-gulis [18, Conjectures H1, H2], namely the Sprindzuk conjecture, can easilybe deduced from Theorem 1. Let us recall this result. For q ∈ Z d we define Π + ( q ) := (cid:81) d | q i | + , where | x | + := max { , | x |} for all x ∈ R . A point y ∈ R n issaid to be very well multiplicatively approximable (or VWMA for short) if forsome ε > there are infinitely many q ∈ Z d such that | p + q · y | · Π + ( q ) ≤ Π + ( q ) − ε . (30)A manifold M ⊂ R d is said to be strongly extremal if Lebesgue almost everypoint on M is not VWMA. Theorem 10 (Kleinbock-Margulis [18]) . Let U ⊂ R n be a connected open setand f , . . . , f d : U → R be real analytic functions, which together with , arelinearly independent over R . Write f = ( f , . . . , f d ) . Then M := { f ( x ) , x ∈ U } is strongly extremal. As often with applications of the subspace theorem, proofs proceed by in-duction on dimension. The induction hypothesis will be as follows: if g =( g , . . . , g d ) is a tuple of linearly independent analytic functions on U and b , . . . , b d ∈ Z d are linearly independent, then for almost every x ∈ U andevery ε > , there are only finitely many solutions v ∈ Z d +1 to the inequalities < d (cid:89) | L i ( v ) | ≤ (cid:107) v (cid:107) ε , (31)where (cid:107) v (cid:107) = max | v i | , L ( v ) = g ( x ) · v and L i ( v ) = b i · v for i ≥ . It isstraightforward that this statement implies Theorem 10, by letting v = ( p, q ) , g = (1 , f ) and b , . . . , b d the standard basis of Z d .26et φ ( x ) be the matrix whose rows are L /g , L , . . . , L d . The linear inde-pendence assumption implies that φ ( x ) ∈ GL d +1 ( R ) on an open subset U (cid:48) ⊂ U ,and without loss of generality we may assume that U (cid:48) = U . The equationsdefining the Plücker closure of φ ( U ) are linear combinations of k × k minors of φ ( x ) . Since those are linear combinations of the g i /g , the Plücker closure of φ ( U ) is the Plücker closure of the set of all matrices in GL d +1 ( R ) whose firstrow is (1 , y ) with y ∈ R d arbitrary and whose other rows are L , . . . , L d . So itis defined over Q .We may thus apply Theorem 1 and conclude that there is a finite number ofproper rational hyperplanes V , such that for almost every x , the large enoughsolutions v to (31) are contained in some V .Now pick one such V , and consider the restriction of L to V . For v ∈ V we can write v = (cid:80) d k i v i for k = ( k , . . . , k d ) ∈ Z d , where v , . . . , v d is a basisof V ∩ Z d +1 . Set L (cid:48) ( k ) := L | V ( v ) = v · g = k · h , where h = ( h , . . . , h d ) and h i = v i · g . And for i ≥ , L (cid:48) i ( k ) := L i | V ( v ) = v · b i = k · c i , where c i = ( b i · v , . . . , b i · v d ) . The L i are linearly independent, so the ( L (cid:48) i ) d have rankat least d on R d , and thus ( c , . . . , c d ) has rank at least d − . Up to reordering,we may assume that c , . . . , c d − are linearly independent. Similarly we see that h , . . . , h d are linearly independent.Finally observe that a solution v ∈ V to (31) yields a k ∈ Z d such that < d (cid:89) | L (cid:48) i ( k ) | (cid:28) (cid:107) k (cid:107) ε . But L (cid:48) d ( k ) = v · b d ∈ Z \ { } . Thus < (cid:81) d − | L (cid:48) i ( k ) | (cid:28) (cid:107) k (cid:107) ε and thus, byinduction hypothesis, k belongs to a finite set of points. Hence so does v . Thisends the proof. Ridout’s theorem [5, 39] is an extension of Roth’s theorem where p -adic placesare allowed. This improves the exponent in Roth’s theorem from to in casethe rational approximations have denominators with prime factorization in afixed subset. In this paragraph, we present one possible similar variant of theKleinbock-Margulis theorem (Theorem 10). This will rely on the S -arithmeticsubspace theorem for manifolds (Theorem 6). Theorem 11.
Let S be a finite set of primes. Let U ⊂ R n be a connected opensubset and f , . . . , f d : U → R be real analytic functions, which together with are linearly independent over R . Write f = ( f , . . . , f d ) . Then for every ε > and for Lebesgue almost every u ∈ U , we have max ≤ i ≤ d | qf i ( u ) − p i | ≥ q − ε (32) for all ( p , . . . , p d , q ) ∈ Z d +1 with all prime factors of q in S , except for finitelymany exceptions. q , where the right-hand side of (32) needs to be replaced by theweaker bound q − d − ε . We have chosen simultaneous approximation for a change,but a similar statement with similar proof holds also for linear forms. Besidesthe theorem holds under the weaker assumption that the subspace of R d +1 spanned by all vectors (1 , f ( x ) , . . . , f d ( x )) , x ∈ U , is defined over Q . Proof sketch.
The proof is a straightforward modification of the one we gave ofTheorem 10 in §3.1. We define linear forms on R d +1 , L ( ∞ ) u,i is x if i = 1 and x f i ( u ) − x i if i > . And if p ∈ S , L ( p ) u,i is constant equal to x i . And we notethat if x := ( q, p , . . . , p d ) ∈ Z d +1 contradicts (32) , then (cid:89) ≤ i ≤ d +1 (cid:89) p ∈{∞}∪ S | L ( p ) u,i ( x ) | p ≤ H ( x ) − dε . So Theorem 6 applies and if q is large enough, x belongs to a finite family ofproper rational subspaces. This allows us to use induction after restricting thelinear forms to one of these subspaces, as in the proof of Theorem 10. The proofis left to the reader. In this paragraph, we describe the extension of the Kleinbock-Margulis theoremto the case of subsmanifolds of matrices and we show how to recover fromthe subspace theorem for manifolds one of the main result of [2], which is acriterion for extremality in terms of so-called constraining pencils and an explicitcomputation of the exponent.In what follows, E and V are two finite-dimensional real vector spaces witha Q -structure. We fix a lattice ∆ in V , which defines the rational structure. For x in Hom(
V, E ) , we define β ( x ) = inf { β > | ∃ c > ∀ v ∈ ∆ \ { } , (cid:107) x ( v ) (cid:107) ≥ c (cid:107) v (cid:107) − β } . Note that β ( x ) only depends on the subspace ker x in V .Given a subspace W in V and an integer r , we define the pencil of endomor-phisms P W,r by P W,r = { x ∈ Hom(
V, E ) | dim x ( W ) ≤ r } . We say that P W,r is constraining if dim
W/r < dim
V / dim E and rational if W is so. In the case of algebraic sets defined over Q , the following theorem wasproved in joint work with Menny Aka and Lior Rosenzweig [2, Theorem 1.2].The approach taken here yields a different proof of that result, and allows togeneralize it to subsets defined over Q . Theorem 12 (Diophantine exponent for submanifolds of matrices) . Let U ⊂ R n a connected open set and φ : U → Hom(
V, E ) an analytic map. Assume that he Zariski closure of M = φ ( U ) is defined over Q . Then, for Lebesgue almostevery u in U , setting x = φ ( u ) , β ( x ) = max (cid:26) dim Wr − W rational subspace such that M ⊂ P W,r (cid:27) . Remark.
For any sublattice ∆ (cid:48) ≤ ∆ , one may define r (∆ (cid:48) ) = max { dim Span R x (∆ (cid:48) ); x ∈ M } . Then, the formula in the above theorem is simply, for almost every x in M , β ( x ) = β := max (cid:26) rk ∆ (cid:48) r (∆ (cid:48) ) − (cid:48) ≤ ∆ (cid:27) . Proof of Theorem 12.
We prove the theorem by induction on d = dim V , usingthe subspace theorem. d = 1 . The result is clear because, for every x , the subgroup x (∆) is a discretesubgroup of E . d − → d . Suppose the result has been proven for d − ≥ . Let m = dim E and d = dim V . Fixing bases for E and ∆ , we identify Hom(
V, E ) with m × d matrices. Given x in M , we denote by x i , ≤ i ≤ d , its columns, whichare vectors in E . The rank of x is almost everywhere constant, and taking acoordinate projection if necessary, we assume that this rank is m ≤ d and thatfor almost every x in M , the vector space E is spanned by the first m columns x i , ≤ i ≤ m . We want to show that, for almost every x in M , for all ε > ,the set of inequalities | v x i + v x i + · · · + v d x id | ≤ (cid:107) v (cid:107) − β − (cid:15) , ≤ i ≤ m, (33)has only finitely many solutions v = ( v , . . . , v d ) in Z d . For ≤ i ≤ m , define alinear form on V (cid:39) R d by L i ( v ) = v x i + v x i + · · · + v d x id , and for m < i ≤ d , L i ( v ) = v i . Since ( x i ) ≤ i ≤ m spans E , the family of linear forms ( L i ) ≤ i ≤ d is linearly inde-pendent. Moreover, as β ≥ rk ∆ r (∆) − d − mm , condition (33) certainly implies d (cid:89) i =1 | L i ( v ) | ≤ (cid:107) v (cid:107) − (cid:15) . We may therefore apply Theorem 1: there exists a finite family V , . . . , V h ofhyperplanes in Q k such that, for almost every x in M , the integer solutions to(33) all lie in the union V ∪ · · · ∪ V h except a finite number of them. It nowsuffices to check that in each V i , there can be only finitely many solutions. Thisfollows from the induction hypothesis applied to V (cid:48) = V i , ∆ (cid:48) = V i ∩ ∆ and tothe manifold M (cid:48) image of M under restriction to V (cid:48) .The converse inequality β ( x ) ≥ β is true for all x in M , as is easily seenusing the classical Dirichlet argument.29t is also worth observing the following relation between the notions of ex-tremality and semistability. In [18, 19] an analytic submanifold M of M m,n ( R ) is said to be extremal if for almost every Y ∈ M and every ε > there are onlyfinitely many vectors q ∈ Z n and p ∈ Z m such that (cid:107) Y q − p (cid:107) ≤ (cid:107) q (cid:107) − ( n/m + ε ) .Now consider the matrix L Y := (cid:18) I m YI n (cid:19) The image (cid:102) M of M under Y (cid:55)→ L Y defines a submanifold of GL n + m ( R ) . UsingProposition 2 and Theorem 3, it is then easy to see that: Proposition 5 (extremality vs. semistability) . If R n + m is (cid:102) M -semistable withrespect to the unimodular flow: a t = ( e tn , . . . , e tn , e − tm , . . . , e − tm ) , then M is extremal. If the Zariski (or Plücker) closure of M is defined over Q ,then the almost sure diophantine exponent β from Theorem 12 can be read offthe rational Grayson polygon of (cid:102) M by the formula β = n + mγ + m , where γ is the smallest slope of the rational Grayson polygon. In particular Q n + m is (cid:102) M -semistable if and only if M is extremal. Following the suggestion of Baker [3, page 96] to study the multiplicative dio-phantine properties of the Mahler curve, Kleinbock and Margulis introducedthe notion of strong extremality for manifolds in R n . This was later generalizedin [19, 4] to the context of diophantine approximation on matrices, but in thatgeneralized setting the optimal criterion for strong extremality remained to befound [19, 4]. The method of the present paper can be used to answer thisproblem. Below we apply Theorem 1 and give a complete solution when theZariski closure of the manifold is defined over Q .Let m and n be two positive integers, and M m,n ( R ) the space of m × n matrices with real entries. Following Kleinbock and Margulis [18], we say thata matrix Y ∈ M m,n ( R ) is very well multiplicatively approximable (VWMA) ifthere exists ε > such that the inequality m (cid:89) i =1 | Y i q − p i | ≤ n (cid:89) j =1 | q j | − − ε + has infinitely many solutions ( p , q ) ∈ Z m × Z n . In the above inequality, Y i denotes the i -th row of Y , for i = 1 , . . . , m , and | q | + = max( | q | , . More30enerally, we define as in [8, §1.4] the multiplicative diophantine exponent of Y ∈ M m,n ( R ) as ω × ( Y ) = sup ω > m (cid:89) i =1 | Y i q − p i | ≤ n (cid:89) j =1 | q j | − ω + for infinitely many ( p , q ) . With this definition, we see that a matrix Y is VWMA if and only if ω × ( Y ) > .Using the Borel-Cantelli lemma, it is not difficult to check that for the Lebesguemeasure, almost every Y in M m,n ( R ) satisfies ω × ( Y ) = 1 . It is therefore naturalto ask what other measures µ on M m,n ( R ) satisfy this property.We now set up some notation to formulate our criterion for stong extremality.Given a matrix Y in M m,n ( R ) , with rows Y , . . . , Y m , we let L Y = (cid:18) − I Y I (cid:19) , and denote by L i , i = 1 , . . . , m + n the linear forms on R m + n given by the rowsof the matrix L Y : (cid:26) L i ( p , q ) = Y i q − p i for i ∈ { , . . . , m } L i ( p , q ) = q i − m for i ∈ { m + 1 , . . . , m + n } . If W is a linear subspace of R n + m , and I a non-empty subset of { , . . . , m + n } ,we let s I,W = rk( L i | W ) i ∈ I . Definition 7 (Multiplicative pencils) . Let
I, J be proper subsets of { , . . . , m + n } such that I ⊂ { , . . . , m } ⊂ J and r, s non-negative integers. Given asubspace W ≤ R n + m , we define a subvariety of endomorphisms P I,J,r,s,W ⊂ M m,n ( R ) by P I,J,r,s,W = { Y ∈ M m,n ( R ) | s I,W ≤ r and s J,W ≤ s } . To justify the relevance of this definition to our problem, we start by an easyproposition, which is a consequence of Minkowski’s first theorem or Dirichlet’spigeonhole principle.
Proposition 6 (Dirichlet’s principle) . Fix Y ∈ M m,n ( R ) , and denote by L i , i = 1 , . . . , m + n , the rows of the matrix L Y . Assume that W is a k -dimensionalrational subspace of R m + n and that ∅ (cid:54) = I ⊂ { , . . . , m } ⊂ J (cid:40) { , . . . , m + n } are such that r = rk( L i | W ) i ∈ I and s = rk( L i | W ) i ∈ J . Then, ω × ( Y ) ≥ ( k − s ) | I | r ( n + m − | J | ) . emark. By convention, if r = 0 , the ratio is equal to + ∞ . Note that one canalways take W = R m + n , I = J = { , . . . , m } , and r = s = m , in which case theratio is equal to . Proof. If r = 0 , then one must have Y i q − p i = 0 for some integer vector ( p , q ) in Z m + n . It is then clear that ω × ( Y ) = ∞ . So we may assume that r (cid:54) = 0 .Since by definition ω × ( Y ) ≥ , we may also assume that s < k , otherwise thereis nothing to prove.Assuming that Y ∈ P I,J,r,s,W , we shall prove that there exists a constant
C > depending only on Y and W such that the inequality m (cid:89) i =1 | L i ( v ) | ≤ C n (cid:89) j =1 | L m + j ( v ) | − ( k − s ) | I | r ( n + m −| J | ) has infinitely many solutions v in W . This will yield the desired lower boundon ω × ( Y ) . Let Q > be a large parameter. Pick i , . . . , i r in I such that L i | W , . . . , L i r | W are linearly independent, pick i r +1 , . . . , i s in J such that L i | W , . . . , L i s | W are linearly independent, and complete with i s +1 , . . . , i k such that L i | W , . . . , L i k | W are linearly independent. The symmetric convex body in W defined by | L i (cid:96) ( v ) | ≤ Q − ( k − s ) for ≤ (cid:96) ≤ r | L i (cid:96) ( v ) | (cid:28) for r < (cid:96) ≤ s | L i (cid:96) ( v ) | ≤ Q r for s < (cid:96) ≤ k has volume (cid:29) and therefore, by Minkowski’s first theorem, it contains a non-zero point v in W ∩ Z n + m . By our choice of the indices i (cid:96) , ≤ (cid:96) ≤ k , such apoint satisfies | L i ( v ) | ≤ Q − ( k − s ) for i ∈ I | L i ( v ) | (cid:28) for i ∈ J \ I | L i ( v ) | ≤ Q r for i (cid:54)∈ J and therefore m (cid:89) i =1 | L i ( v ) | (cid:28) Q −| I | ( k − s ) (cid:28) n (cid:89) j =1 | L m + j ( v ) | − ( k − s ) | I | r ( n + m −| J | ) . Now, as in the introduction, let M = φ ( U ) be a connected analytic subman-ifold of M m,n ( R ) endowed with the measure µ equal to the push-forward of theLebesgue measure under the analytic map φ : U → M m,n ( R ) . When the Zariskiclosure of M is defined over Q – for example when φ is given by a polynomialmap with coefficients in Q – the above proposition actually provides a formulafor ω × ( Y ) , when Y is a µ -generic point of M . In other words: Theorem 13 (Formula for the multiplicative exponent) . Assume that the Zariskiclosure of M is defined over Q . Then for µ -almost every Y in M , ω × ( Y ) = max M ⊂P I,J,r,s,W W rational (dim W − s ) | I | r ( m + n − | J | ) .
32e stated the result for analytic submanifolds for convenience, but it holdswith the same proof for all good measures µ in the sense of §1.9 provided theZariski closure of the support of µ is defined over Q . We shall say that amultiplicative pencil P I,J,r,s,W is constraining if it satisfies (dim W − s ) | I | r ( m + n − | J | ) > . Our criterion for strong extremality immediately follows from the above for-mula. Corollary 3 (Criterion for strong extremality) . If M is an analytic submani-fold of M m,n ( R ) whose Zariski closure is defined over Q . Then M is stronglyextremal if and only if it is not contained in any rational constraining pencil. The proof of Theorem 13 is inspired by Schmidt’s proof of an analogousresult on products of linear forms [33, §12, page 242].
Proof of Theorem 13.
Let ω > be such that for Y in a set of positive measurein M , the inequality m (cid:89) i =1 | Y i q − p i | ≤ m (cid:89) j =1 | q j | − ω + (34)has infinitely many solutions ( p , q ) in Z m + n . We want to show that M isincluded in a pencil P I,J,r,s,W such that (dim W − s ) | I | r ( m + n − | J | ) ≥ ω. Let W be a rational subspace of minimal dimension k containing infinitelymany solutions to (34). If k = 1 , then there must exist i ∈ { , . . . , m } and ( p , q ) ∈ Z m + n such that Y i q − p i = 0 , and one can take I = { i } , J = { , . . . , m } , r = 0 , and s = 1 . So we assume k ≥ . Reordering the indices if necessary, wemay assume that W contains infinitely many solutions to (34) satisfying < | Y q − p | ≤ · · · ≤ | Y m q − p m | (35)and | q | ≤ · · · ≤ | q n | . (36)Define i < i < · · · < i f ≤ m inductively so that each i (cid:96) is minimal such that rk( L i | W , . . . , L i (cid:96) | W ) = (cid:96) , and then m < j < · · · < j g ≤ m + n such that j (cid:96) isminimal such that rk( L i | W , . . . , L i f | W , L j | W , . . . , L j (cid:96) | W ) = f + (cid:96) . Note thatin our notation, f + g = k = dim W .Given a large solution v = ( p , q ) to (34) in W , choose numbers c , . . . , c f such that | L i (cid:96) ( v ) | (cid:16) (cid:107) v (cid:107) − c (cid:96) for (cid:96) = 1 , . . . , f We were informed by David Simmons [9] that he and Tushar Das also obtained a similarcriterion for strong extremality of submanifolds defined over R . d , . . . , d g such that | L j (cid:96) ( v ) | (cid:16) (cid:107) v (cid:107) d (cid:96) for (cid:96) = 1 , . . . , g. By (35), one has c ≥ · · · ≥ c f ≥ . By (36) and the fact that one always has | L j (cid:96) ( v ) | (cid:28) (cid:107) v (cid:107) , one finds ≤ d ≤ · · · ≤ d g ≤ . Moreover, by minimality of W , the subspace theorem applied in W to the set of linear forms L i | W , . . . , L i f | W , L j | W , . . . , L j g | W shows that − c + · · · − c f + d + · · · + d g ≥ . In conclusion, one sees that the k -tuple ( c , . . . , c f , d , . . . , d g ) belongs to the convex polytope ( P ) defined by ( P ) c ≥ · · · ≥ c f ≥ ≤ d ≤ · · · ≤ d g ≤ d + · · · + d g ≥ c + · · · + c f . Now inequality (34) implies c ( i − i ) + · · · + c f ( m − i f ) − ω [ d ( j − m ) + · · · + d g ( m + n − j g )] ≥ . The linear map f ( c , d ) = c ( i − i ) + · · · + c f ( m − i f ) − ω [ d ( j − m ) + · · · + d g ( m + n − j g )] is non-negative on some point in the convex polytope ( P ) , so itmust be non-negative on one of its vertices. The polytope (P) has gf vertices,given by p a,b : (cid:26) d = · · · = d a < d a +1 = · · · = d g = 1 g − ab = c = · · · = c b > c b +1 = · · · = c f = 0 where (cid:26) a = 0 , . . . , g − b = 1 , . . . , f Choose ( a, b ) such that f ( p a,b ) ≥ , and let I = { , . . . , i b } J = { , . . . , j a } r = bs = f + a. We then obtain ≤ f ( p a,b ) = g − ab | I | − ω ( m + n − | J | ) , whence ( k − s ) | I | r ( m + n − | J | ) ≥ ω. Remark.
Strong extremality also relates to semistability in a similar way as inProposition 5. We leave it to the reader to check that in the setting of Theorem13, M is strongly extremal if and only if Q m + n is semistable for { L Y , Y ∈ M } with respect to all unimodular flows a t = ( e tA , . . . , e tA n + m ) with A ≥ . . . ≥ A m ≥ ≥ A m +1 ≥ . . . ≥ A m + n . See [19] where the relevance of this family offlows for strong extremality was uncovered.34 .5 Roth’s theorem for nilpotent Lie groups Diophantine approximation on nilpotent Lie groups was studied in [1, 2]. In thisparagraph we continue this study and derive an analogue of Roth’s theorem [28]in this context. In doing so we extend Theorem 12 to approximation with quasi-norms.
Diophantine approximation in nilpotent Lie groups
In this paragraph G will denote a simply connected nilpotent Lie group, endowedwith a left-invariant riemannian metric. We identify it with its Lie algebra g under the exponential map, so that its Haar measure is given by the Lebesguemeasure on g . For a finite symmetric subset S of G and Γ = (cid:104) S (cid:105) the subgroupit generates, the diophantine exponent of Γ in G is β (Γ) = inf { β | ∃ c > ∀ n ∈ N ∗ , ∀ x ∈ S n \ { } , d ( x, ≥ cn − β } , where S n denotes the set of elements of Γ that can be written as a product ofat most n elements of S . Because Γ is nilpotent, this definition does not dependon the choice of S . The following theorem was proved in [2, Theorem 7.4]. Theorem 14 (Existence of the exponent) . Let G be a connected and simplyconnected real nilpotent Lie group endowed with a left-invariant riemannianmetric d . For each k ≥ , there is β k ∈ [0 , ∞ ] such that for almost every k -tuple g = ( g , . . . , g k ) ∈ G k with respect to the Haar measure, the subgroup Γ g generated by g , . . . , g k satisfies β (Γ g ) = β k . We also showed in [2] that when G is rational, i.e. when its Lie algebra g admits a basis with rational structure constants, then β k can be explicitlycomputed and in particular is rational. In fact β k = F ( k ) for some rationalfunction F ∈ Q ( X ) when k is large enough. Using the subspace theorem formanifolds, Theorem 1, we can now show the following. Theorem 15 (Roth’s theorem for nilpotent groups) . Let G be a connectedsimply connected nilpotent Lie group and k a positive integer. Assume that theLie algebra g of G admits a basis with structure constants in a number field K .Then β k ∈ Q . Moreover, for every k -tuple g ∈ g ( Q ) k we have β (Γ g ) ∈ Q . Furthermore, thereis C = C (dim g ) > and a countable union U of proper algebraic subsets of g k defined over K and of degree at most C , such that β (Γ g ) = β k for every g ∈ g k ( Q ) \ U . k = 2 and G = ( R , +) is exactly Roth’s theorem. In this case β = 1 and U is the family of lines in R with rational slopes.The basic idea for the proof of Theorem 15, developed in [2], is to reducethe problem to a question of diophantine approximation on submanifolds. Forthat, we introduce the free Lie algebra F k over k generators x , . . . , x k . Inside F k , the ideal of laws L k, g on the Lie algebra g of G is the set of elements r in F k such that r ( X , . . . , X k ) = 0 for every X , . . . , X k in g , and the ideal of rationallaws L k, g , Q is the real span of the intersection of L k, g with F k ( Q ) , the natural Q -structure on F k . The Lie algebra F k, g , Q = F k / L k, g , Q has a graded structure F k, g , Q = s (cid:77) i =1 F [ i ] k, g , Q , where F [ i ] k, g , Q is the homogeneous part of F k, g , Q consisting of brackets of degree i . For r = (cid:80) r i with r i ∈ F [ i ] k, g , Q , we let | r | := max i =1 ,...,s (cid:107) r i (cid:107) i , (37)where (cid:107) · (cid:107) is a fixed norm on F k, g , Q . Endowed with this quasi-norm, the Liealgebra F k, g , Q is quasi-isometric to the group of word maps on G , endowed withthe word metric [2, Proposition 7.2]. This yields the following characterizationfor the above diophantine exponent β (Γ g ) proved in [2, Proposition 7.3]. Wesay that Γ g is relatively free in G if the only relations satisfied by g are thelaws of G . This holds for all g outside a countable union of proper algebraicsubvarieties of bounded degree defined over Q , and in particular for Lebesguealmost every g ∈ G k . Proposition 7.
Let G be a simply connected nilpotent Lie group, with Liealgebra g . Let g = ( e X , . . . , e X k ) be a k -tuple in G such that Γ g is relativelyfree in G . Then the exponent β (Γ g ) defined above is also the infimum of all β > such that (cid:107) r ( X , . . . , X k ) (cid:107) ≥ | r | − β (38) holds for all but finitely many r ∈ F k, g , Q ( Z ) . In the next paragraph, we show how Theorem 1 yields a formula for dio-phantine exponents defined with quasi-norms, as in the above proposition.
Weighted diophantine approximation
This paragraph generalizes the results of §3.3 to diophantine approximation withquasi-norms, also called weighted diophantine approximation (see e.g. [20, §1.5]and references therein). Again, E and V are two finite-dimensional real vectorspaces, and ∆ is a lattice in V , defining a rational structure.We fix a norm (cid:107) · (cid:107) on E . On V , we measure the size of vectors using aquasi-norm | · | given by the formula | v | = max ≤ i ≤ d |(cid:104) v, u ∗ i (cid:105)| αi , (39)36here α = ( α , . . . , α d ) is a d -tuple of positive real numbers and ( u ∗ i ) ≤ i ≤ d abasis of V ∗ . Given x in Hom(
V, E ) , define the diophantine exponent of x by β α ( x ) = inf { β > | ∃ c > ∀ v ∈ ∆ , (cid:107) xv (cid:107) ≥ c | v | − β } . Definition 8 (Growth rate for a quasi-norm) . Let W be a linear subspace in V . The growth rate of balls in W for the quasi-norm | · | is given by α ( W ) = lim R →∞ R log Vol { v ∈ W | | v | ≤ R } . Remark.
It is not hard to see that this limit exists and equals (cid:80) i ∈ I W α i for acertain subset I W ⊂ [1 , d ] . Indeed the restriction of | · | to W is itself comparableup to multiplicative constants to a quasi-norm with exponents α i , i ∈ I W ,where I W = { i , . . . , i k } is defined as follows. Choose i minimal such that therestriction of u ∗ i to W is non-zero, then inductively choose i j minimal such thatthe linear forms u ∗ i | W , . . . , u ∗ i j | W are linearly independent.This gives us a lower bound for the diophantine exponent, using a standardDirichlet type argument: Lemma 5.
Let V and | · | be as above. For any x in Hom(
V, E ) , β α ( x ) ≥ max { α ( W ∩ ker x )dim W − dim W ∩ ker x ; W ≤ V rational subspace } . Proof.
Let
R > be some large parameter. The number of points v in W ∩ ∆ such that | v | ≤ R is roughly R α ( W ) , and their images in x ( W ) (cid:39) W/ ( W ∩ ker x ) lie in a distorted ball of volume O ( R α ( W ) − α ( W ∩ ker x ) ) . Comparing volumes,we find that balls of radius ε around those points cannot be disjoint if ε (cid:29) R − α ( W ∩ ker x )dim W − dim W ∩ ker x .It turns out that this lower bound is in fact attained almost everywhereon analytic submanifolds of Hom(
V, E ) whose Zariski closure is defined over Q . This is the content of the next theorem. As earlier, we endow M := φ ( U ) with the push-forward µ of the Lebesgue measure on the connected open subset U ⊂ R d via the analytic map φ : U → Hom(
V, E ) . Theorem 16 (Diophantine exponent for quasi-norms) . Assume V and | · | areas above, and that the Zariski closure Zar ( M ) of M is defined over Q . Then,for almost every x in M , β α ( x ) = max (cid:26) min y ∈ M α ( W ∩ ker y )dim W − dim W ∩ ker y ; W ≤ V rational subspace (cid:27) . (40) This equality is also true for every Q -point of Zar ( M ) outside a union of properalgebraic subsets of M defined over Q and of bounded degree. Remark.
Note that α can take only finitely many values, so the maximum andminimum in the above formula are indeed attained.37 emark. The theorem implies that β α ( x ) ≥ β for almost every x on M if andonly if β α ( y ) ≥ β for every y on M , if and only if there exists a rational subspace W such that M is included in the (closed) algebraic subset of all y ∈ Hom(
V, E ) such that α ( W ∩ ker y ) ≥ β (dim W − dim W ∩ ker y ) . Proof of Theorem 16.
Since we can always reorder the u ∗ i , ≤ i ≤ d , we mayassume without loss of generality that < α ≤ · · · ≤ α d . Then, for x in Hom(
V, E ) , let m = dim x ( V ) and n = dim ker x. and define inductively a subset I x = { i ,x , . . . , i n,x } in { , . . . , d } byfor j = 1 , i ,x = min { i | u ∗ i | ker x (cid:54) = 0 }∀ j ≥ , i j +1 ,x = min { i | ( u ∗ i ,x | ker x , . . . , u ∗ i j,x | ker x , u ∗ i | ker x ) is linearly independent } . Now suppose L x is an element of GL( d, R ) with rows L j,x satisfying1. ker x = (cid:84) ≤ j ≤ m ker L j,x ∀ j ∈ { , . . . , n } , L m + j,x = u ∗ i j,x Then β α ( x ) is the minimal β > such that, for all ε > , for Q large enough,the set of inequalities (cid:26) | L j,x ( v ) | ≤ Q − β − ε for ≤ i ≤ m | L m + j,x ( v ) | ≤ Q α ij for ≤ j ≤ n (41)has no non-zero solution v in ∆ .Let β be the right-hand side of (40) . We want to show that for µ -almostevery x , for all ε > and for Q > large enough, the inequalities (41) have nonon-zero solution in ∆ . We prove this by induction on d , using our subspacetheorem (i.e. Theorem 1). d = 1 : There is not much to prove in this case, since ker x is equal { } or V , for µ -almost every x . d − → d : Assume the result already proven for d − . The map x (cid:55)→ I x is con-stant on a dense Zariski open subset of Zar ( M ) ; we denote by I = { i , . . . , i n } its value on this dense open subset, and by (cid:102) M the set of all L x , for x in thatsubset and in M . Since β ≥ α ( V ) m = (cid:80) nj =1 α ij m , the solutions v to (41) satisfy d (cid:89) i =1 | L i,x ( v ) | ≤ Q − mβ − mε + (cid:80) nj =1 α ij ≤ (cid:107) v (cid:107) − ε (cid:48) , and therefore, by Theorem 1, there exist proper subspaces V , . . . , V (cid:96) in Q d containing all solutions except a finite number. All we need to check is that onin each V i , there can be only finitely many solutions. But this follows from theinduction hypothesis, applied to V (cid:48) = V i , ∆ (cid:48) = ∆ ∩ V i , |·| = |·| V (cid:48) (the restrictionof a quasi-norm to a subspace is comparable to a quasi-norm [2, §4.1]), and tothe submanifold { x | V i ; x ∈ (cid:102) M } . We leave it to the reader to check from theformula defining the exponent that β (cid:48) ≤ β .38e can now easily derive our theorem about nilpotent groups. Proof of Theorem 15.
Let V = F k, g , Q , ∆ = F k, g , Q ( Z ) and E = g . For each k -tuple g = ( e X , . . . , e X k ) in G k , we obtain an element x g in Hom(
V, E ) givenby evaluation at ( X , . . . , X k ) : x g ( r ) = r ( X , . . . , X k ) . The map G k → Hom(
V, E ) , g (cid:55)→ x g is a polynomial map with coefficients in K . In particular the Zariski closure of its image is defined over Q . When Γ g is relatively free, Proposition 7 shows that β (Γ g ) = β ( x g ) , where β ( x g ) is thediophantine exponent with respect to the quasi-norm | · | on V defined in (37) .By Theorem 16, β ( x g ) = max (cid:26) min h ∈ G k α ( W ∩ ker x h )dim W − dim W ∩ ker x h ; W ≤ V rational subspace (cid:27) for Lebesgue almost every g ∈ G k . This shows that β k is well defined, andsince α takes rational values, this formula shows that β k ∈ Q . For each rational W ≤ V , the set of all h ∈ G k such that α ( W ∩ ker x h ) > β k dim( W/ ( W ∩ ker x h ) is a proper algebraic subset of G k defined by equations of bounded degree withcoefficients in K . Their union forms a proper subset of G k by Lemma 4 andTheorem 16 implies that β (Γ g ) = β k for every g outside this union. Acknowledgements:
We are grateful to Barak Weiss for stimulating conversa-tions about Harder-Narasimhan filtrations and to Mohammed Bardestani, Ves-selin Dimitrov, Elon Lindenstrauss and Péter Varjú for interesting discussionsaround the subspace theorem.
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