DIRICHLET’S THEOREM ON DIOPHANTINE
APPROXIMATION AND HOMOGENEOUS FLOWS
DMITRY KLEINBOCK AND BARAK WEISS
Preliminary as of May 25, 2006
1. Introduction
Let m, n be positive integers, and denote by Mm,n the space of m × n
matrices with real entries. Dirichlet’s Theorem (hereafter abbreviated
by ‘DT’) on simultaneous diophantine approximation states that for
any Y ∈ Mm,n (viewed as a system of m linear forms in n variables)
and for any t > 0 there exist q = (q1 , . . . , qn ) ∈ Zn r {0} and p =
(p1 , . . . , pm ) ∈ Zm satisfying the following system of inequalities:
kY q − pk < e−t/m
and kqk ≤ et/n .
(1.1)
Here and hereafter (unless otherwise specified) k · k stands for the norm
on Rk given by kxk = max1≤i≤k |xi |. See [Sc] for a discussion of two
ways of proving this theorem, due to Dirichlet and Minkowski respectively.
Given Y as above and positive ε < 1, we will say that DT can be
ε-improved for Y , and write Y ∈ DIε (m, n), or Y ∈ DIε when the
dimensionality is clear from the context, if for every sufficiently large t
one can find q ∈ Zn r {0} and p ∈ Zm with
kY q − pk < εe−t/m
and kqk < εet/n ,
(1.2)
that is, satisfy (1.1) with right hand sides multiplied by ε (for convenience we will also replace ≤ in the second inequality by <). Also
note that Y is called singular if Y ∈ DIε for any ε > 0 (in other words,
DT can be ‘infinitely improved’ for Y ).
The two papers [DS1, DS2] by H. Davenport and W. Schmidt give a
few basic results concerning the properties defined above. For example,
the following is proved there:
Theorem 1.1 ([DS2]). For any1 m, n ∈ N and any ε < 1, the sets
DIε (m, n) have Lebesgue measure zero.
1Even
though the results of [DS2] are stated for matrices with one row or one
column, i.e. for a vector or a single linear form, the proofs can be generalized to
the setting of systems of linear forms.
1
2
DMITRY KLEINBOCK AND BARAK WEISS
In other words, λ-generic systems of linear forms do not allow any
improvement to DT (λ will denote Lebesgue measure throughout the
paper).
Another question raised by Davenport and Schmidt concerns a possibility (or impossibility) of improving DT for matrices with some functional relationship between entries. Specifically they considered row
matrices
f (x) = (x, x2 ) ∈ M1,2 ,
(1.3)
and proved
Theorem 1.2 ([DS1]). λ x ∈ R : f (x) ∈ DIε = 0 ∀ ε < 4−1/3 .
In other words, generic matrices of the form f (x) do not allow a sufficiently drastic improvement to DT. This result has been later extended
by R. Baker, Y. Bugeaud, and others. Namely, for some other smooth
submanifolds of Rn they have exhibited constants ε0 such that almost
no points on these submanifolds (viewed as row or column matrices)
are in DIε for ε < ε0 . We will discuss the history in more detail in §4.
In the present paper we significantly generalize Theorems 1.1 and
1.2 by using a homogeneous dynamics approach and following a theme
developed in the paper [KW2] which dealt with infinite improvement
to DT, that is, with singular systems. Namely, we study the subject of
improvement of the multiplicative version of DT. In what follows we
fix m, n ∈ N and let k = m + n. Let us denote by a+ the set of k-tuples
t = (t1 , . . . , tk ) ∈ Rk with
t1 , . . . , tm > 0, tm+1 , . . . , tk < 0,
and
k
X
ti = 0 .
(1.4)
i=1
It is not hard to see that both Dirichlet’s and Minkowski’s proofs of
DT can easily yield the following statement:
Theorem 1.3. For any system of m linear forms Y1 , . . . , Ym (rows of
Y ∈ Mm,n ) in n variables and for any t ∈ a+ there exist solutions
q = (q1 , . . . , qn ) ∈ Zn r {0} and p = (p1 , . . . , pm ) ∈ Zm of
(
|Yi q − pi | < e−ti ,
i = 1, . . . , m
(1.5)
−tm+j
|qj | ≤ e
,
j = 1, . . . , n .
Now, given an unbounded subset T of a+ and positive ε < 1, say
that DT can be ε-improved for Y along T , or Y ∈ DIε (T ), if for every
t = (t1 , . . . , tk ) ∈ T with large enough norm, the inequalities
(
|Yi q − pi | < εe−ti ,
i = 1, . . . , m
(1.6)
−tm+j
|qj | < εe
,
j = 1, . . . , n .
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
3
i.e., (1.5) with right hand sides multiplied by ε, have nontrivial integer
solutions. Clearly DIε = DIε (R) where
def t
R=
, . . . , mt , − nt , . . . , − nt : t > 0
(1.7)
m
is the ‘central ray’ in a+ . Also say that Y is singular along T if it
belongs to DIε (T ) for every positive ε. The latter definition was introduced in [KW2] for T contained in an arbitrary ray in a+ emanating
from the origin (the set-up of diophantine approximation with weights)
in the special case n = 1.
To state our first result we need one more definition. Denote
def
btc = min |ti | ,
i=1,...,k
+
and say that T ⊂ a drifts away from walls if
∀s > 0 ∃t > 0 :
t ∈ T , ktk > t ⇒ btc > s .
(1.8)
In other words, each coordinate of t tends to infinity as t → ∞ in T .
Using Lebesgue’s Density Theorem and an elementary argument
which can be found e.g. in [Ca, Chapter V, §7] and dates back to
Khintchine, one can show that for any m, n and T ⊂ a+ drifting away
from walls, DIε (T ) has Lebesgue measure zero as long as ε < 1/2.
However proving the same for any ε < 1 is more difficult, even in the
case T = R considered by Davenport and Schmidt. In [DS2] they
derived this fact from the density of generic trajectories of certain oneparameter subgroup action on the space G/Γ, where
G = SLk (R) and Γ = SLk (Z) .
(1.9)
In §2 we show how the fact that DIε (T ) has Lebesgue measure zero
for any ε < 1 and any unbounded T ⊂ R follows from mixing of
the G-action on G/Γ. We also explain why mixing is not enough to
obtain such a result for an arbitrary T ⊂ a+ drifting away from walls,
even contained in a single ‘non-central’ ray, and prove the following
multiplicative analogue of Theorem 1.1:
Theorem 1.4. For any T ⊂ a+ drifting away from walls and any
ε < 1, the set DIε (T ) has Lebesgue measure zero.
In other words, given any T as above, DT in its generalized form
(Theorem 1.3) cannot be improved for generic systems of linear forms.
Theorem 1.4 is derived from the equidistribution of translates of certain
measures on G/Γ (Theorem 2.2). The proof of this theorem, described
in §2, is a modification of arguments used in analyzing epimorphic
groups [We, SW]. It relies on M. Ratner’s classification of measures
invariant under unipotent subgroups, and the ‘linearization method’
4
DMITRY KLEINBOCK AND BARAK WEISS
developed by S. G. Dani, G. A. Margulis, N. Shah and others. An alternative approach avoiding the use of Ratner’s theorem was described
to the authors by Margulis.
Next, building on the approach of [KLW, KW2], we generalize Theorem 1.2, namely, consider measures other than Lebesgue. Here we will
restrict ourselves to measures µ on Rn ∼
= M1,n and study diophantine
n
properties of µ-almost all y ∈ R interpreted as row vectors (linear
forms); that is, we put m = 1 and k = n + 1. Note that the dual case
of simultaneous approximation (n = 1, which was the set-up of [KLW]
and [KW2]) can be treated along the same lines.
All measures on Euclidean spaces will be assumed to be Radon (locally finite regular Borel). Generalizing the set-up of Theorem 1.2, we
will consider measures on Rn of the form f∗ ν, where ν is a measure on
Rd and f a map from Rd to Rn . Our assumptions on f and ν rely on
definitions of:
• measures ν which are D-Federer on open subsets U of Rd , and
• functions f : U → R which are (C, α)-good on U w.r.t. ν
(here D, C, α are positive constants). These definitions can be found
in §3.2, as well as in in [KM2, KLW] and other recent papers dealing with applications of homogeneous dynamics to metric diophantine
approximation.
Now given a measure ν on Rd , an open U ⊂ Rd with ν(U ) > 0 and
a map f : Rd → Rn , say that a pair (f , ν) is
• (C, α)-good on U if any linear combination of 1, f1 , . . . , fn is
(C, α)-good on U with respect to ν;
• nonplanar on U if for any ball B ⊂ U centered in supp ν, the
restrictions of 1, f1 , . . . , fn to B ∩ supp ν are linearly independent over R; in other words, if f (B ∩ supp ν) is not contained
in any proper affine subspace of Rn ;
• (C, α)-good (resp., nonplanar) if for ν-a.e. x there exists a
neighborhood U of x such that ν is (C, α)-good (resp., nonplanar) on U .
Similarly, we will say that ν is D-Federer if for ν-a.e. x ∈ Rd there
exists a neighborhood U of x such that ν is D-Federer on U .
In §3 we prove
Theorem 1.5. For any d, n ∈ N and C, α, D > 0 there exists ε0 =
ε0 (d, n, C, α, D) with the following property. Let ν be a measure on Rd
and f a continuous map from open U ⊂ Rd with ν(U ) > 0 to Rn .
Assume that ν is D-Federer, and (f , ν) is (C, α)-good and nonplanar.
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
Then for any ε < ε0
f∗ ν DIε (T ) = 0 ∀ T ⊂ a+ drifting away from walls.
5
(1.10)
One can easily check, see §4.1, that (f , λ) as in (1.3) satisfies the
assumptions of the above theorem. Moreover, Theorem 1.5 generalizes
all known results on improvement of DT on manifolds, and is applicable to many other examples of interesting measures, see §4.3 for more
examples. The proof is based on so-called ‘quantitative nondivergence’
technique, a generalization of non-divergence of unipotent flows on homogeneous spaces first established by Margulis in early 1970s.
Note that ε0 in the above theorem can be explicitly computed in
terms of d, n, C, α, D, although the value that can be obtained seems
to be far from optimal. For example, we are very far from improving the
values of ε0 obtained in [DS1, DRV, Bu]. On the other hand the set-up
of Theorem 1.5 is much more general than that of the aforementioned
papers, as illustrated by examples considered in §4.
Acknowledgements: This research was supported by BSF grant
2000247 and NSF grant DMS-0239463. Thanks are due to Roger Baker
and Yann Bugeaud for useful discussions.
2. Theorem 1.4 and equidistribution
2.1. The correspondence. Let G and Γ be as in (1.9), and denote
by π the quotient map from G onto G/Γ. G acts on G/Γ by left
translations via the rule gπ(h) = π(gh), g, h ∈ G. Define
Im Y
def
def
τ (Y ) =
, τ̄ = π◦τ ,
0 In
where I` stands for the ` × ` identity matrix. Since Γ is the stabilizer
of Zk under the action of G on the set of lattices in Rk , G/Γ can be
identified with GZk , that is, with the set of all unimodular lattices
in Rk . To highlight the relevance of the objects defined above to the
diophantine problems considered in the introduction, note that
Yq−p
m
n
τ̄ (Y ) =
:p∈Z , q∈Z
.
(2.1)
q
Now for ε > 0 let
def
Kε = π
g ∈ G : kgvk ≥ ε ∀ v ∈ Zk r {0} ,
(2.2)
i.e., Kε is the collection of all unimodular lattices in Rk which contain
no nonzero vector of norm smaller than ε. By Mahler’s compactness
criterion (see e.g. [Ra, Chapter 10]), each Kε is compact, and for each
6
DMITRY KLEINBOCK AND BARAK WEISS
compact K ⊂ G/Γ there is ε > 0 such that K ⊂ Kε . Note also that Kε
is empty if ε > 1 by Minkowski’s Lemma, and has nonempty interior
if ε < 1.
Now to any t ∈ a+ let us associate the diagonal matrix
def
gt = diag(et1 , . . . , etk ) ∈ G .
Then, using (2.1), it is straightforward to see that the system (1.6) has
a nonzero integer solution if and only if gt τ̄ (Y ) ∈
/ Kε . We therefore
arrive at
Proposition 2.1. For Y ∈ Mm,n , 0 < ε < 1 and unbounded T ⊂ a+ ,
one has Y ∈ DIε (T ) if and only if gt τ̄ (Y ) is outside of Kε for all t ∈ T
with large enough norm. Equivalently,
[
\
DIε (T ) =
{Y : gt τ̄ (Y ) ∈
/ Kε } .
(2.3)
t>0
t∈T , ktk>t
In particular, Y is singular along T if and only if the trajectory
{gt τ̄ (Y ) : t ∈ T } is divergent (i.e. eventually leaves Kε for any ε > 0).
The latter observation was first made in [Da, Proposition 2.12] for
T = R, and then in [Kl1, Theorem 7.4] for an arbitrary ray in a+ .
2.2. Lebesgue measure and uniform distribution. We recall that
one of the goals of this paper is to show that Lebesgue-a.e. Y ∈ Mm,n
does not belong to DIε (T ) whenever ε < 1 and T ∈ a+ drifts away
from walls (Theorem 1.4). It follows from the above proposition that
any Y for which
the set {t ∈ T : gt τ̄ (Y ) ∈ Kε } is unbounded
(2.4)
belongs to DIε (T ). Thus to prove Theorem 1.4 it would suffice to show
that (2.4) holds for Lebesgue–a.e. Y ∈ Mm,n whenever T drifts away
from walls. In Section 2.4 we are going to prove a stronger and more
general statement. Here and hereafter vol stands for the G-invariant
probability measure on G/Γ.
Theorem 2.2. For any x ∈ G/Γ, any continuous compactly supported
function ϕ on G/Γ, any B ⊂ Mm,n with positive and finite Lebesgue
measure, and any T ⊂ a+ drifting away from walls, one has
Z
Z
1
ϕ gt τ (Y )x dλ(Y ) =
ϕ dvol .
(2.5)
lim
t→∞ , t∈T λ(B) B
G/Γ
Furthermore, the convergence in (2.5) is uniform in x when the latter
is restricted to a compact subset of G/Γ.
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
7
1
In other words, gt -translates of the pushforward of λ(B)
λ|B by the
map Y 7→ τ (Y )x weak-∗ converge to vol as t → ∞ in T . By taking
x to be the standard lattice Zk ∈ G/Γ, and approximating Kε (which
has nonempty interior and boundary of measure zero for any ε < 1) by
continuous functions on G/Γ, from Theorem 2.2 one can conclude that
λ {Y ∈ B : gt τ̄ (Y ) ∈ Kε } → λ(B)vol(Kε ) as t → ∞, t ∈ T .
This immediately rules out the existence of a subset B of Mm,n of
positive Lebesgue measure such that (2.4) does not hold for Y ∈ B.
Thus, in view of the remark made in the beginning of §2.2, Theorem
1.4 follows from Theorem 2.2.
Note that the conclusion of Theorem 2.2 is not new for T being a
subset of the ‘central ray’ R, see (1.7). Indeed, one can immediately
see that the group
def
H = τ (Mm,n )
(2.6)
is expanding horospherical (see e.g. [KSS, Chapter 1] for the definition)
with respect to gt for any t ∈ R. Hence Theorem 2.2 in this case follows
from mixing of the G-action on G/Γ. The argument is described e.g.
in [KM1, Proposition 2.1] or [Kl2, Proposition 2.4], and dates back to
the early work of Margulis . On the other hand, H is strictly contained
in the expanding horospherical subgroup relative to gt for any t ∈
a+ r R, thus the argument based on mixing is not applicable in more
general situations. Below we bypass this difficulty using some results
and methods from [We] and [SW].
2.3. Drifting from walls and representations. A fundamental result of M. Ratner (see [Mo] or [KSS, Chapter 3] for a detailed account)
classifies measures on G/Γ invariant under unipotent flows2. In utilizing this theorem, it has proved fruitful to analyze the action of G and
its subgroups on linear subspaces of the Lie algebra of G. Namely, let
g = Lie(G), and let
dim
G−1 ^
d
M
V =
g.
(2.7)
d=1
1
We denote by ρ : G → GL(V ) the natural action of G on V through the
dth exterior powers of the adjoint representation. Given a subgroup H
of G, we will write
V H = {v ∈ V : ∀ h ∈ H, ρ(h)v = v} .
2We
only discuss G, Γ as in (1.9), but Ratner’s results hold in much greater
generality
8
DMITRY KLEINBOCK AND BARAK WEISS
We are going to take H as in (2.6), that is, the unipotent subgroup of
G whose Lie algebra h is spanned by matrices Ei,j with i ∈ {1, . . . , m}
and j ∈ {m + 1, . . . , m + n}. Here Ei,j denotes the matrix with 1 in
the (i, j)-th entry and 0 elsewhere. Also let us denote by A the group
of positive diagonal matrices in G.
The following lemma makes it possible to exploit the ‘drifting away
from walls’ condition imposed in Theorem 2.2:
Lemma 2.3. If T ⊂ a+ drifts away from walls, then for any v ∈
V H r {0} one has
ρ(gt )v → ∞ as t → ∞ , t ∈ T .
Proof. We will first specify a basis for V . Let
def
B1 = {Ei,j : i 6= j},
def
B2 = {Ei,i − Ei+1,i+1 : i = 1, . . . , m + n − 1} ,
and note that
def
{x1 , . . . , xr } = B1 ∪ B2 ,
is a basis of g consisting of joint eigenvectors for Ad(A). Consequently,
{x`1 ∧ · · · ∧ x`d : 1 ≤ `1 < · · · < `d ≤ dim G, 1 ≤ d ≤ dim G − 1}
is a basis of V consisting of joint eigenvectors for ρ(A).
Since A normalizes H, the space V H is ρ(A)-invariant, so we can
choose a basis for V H consisting of simultaneous eigenvectors for ρ(A).
There is no loss of generality in assuming that v is such an eigenvector,
i.e. for some d we have
d
^
g.
v = x `1 ∧ · · · ∧ x `d ∈
1
Let p ∈ {1, . . . , d} be such that x`1 , . . . , x`p ∈ B1 , x`p+1 , . . . , x`d ∈ B2 ,
and for q = 1, . . . , p write
x`q = Eiq ,jq .
Note that we have
def
ρ(gt )v = eχ(t) v, where χ(t) =
p
X
(tiq − tjq ) .
(2.8)
q=1
For each i0 ∈ {1, . . . , m + n}, let
def J (i0 ) = r ∈ {1, . . . , m + n} : ∃ q ∈ {1, . . . , p} s.t. iq = i0 , jq = r .
Similarly, for each j0 ∈ {1, . . . , m + n}, let
def I(j0 ) = r ∈ {1, . . . , m + n} : ∃ q ∈ {1, . . . , p} s.t. iq = r, jq = j0 .
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
9
We claim:
J (i0 ) 6⊂ {m + 1, . . . , m + n} =⇒ {m + 1, . . . , m + n} r {i0 } ⊂ J (i0 )
(2.9)
and
I(j0 ) 6⊂ {1, . . . , m}
{1, . . . , m} r {j0 } ⊂ I(j0 ) .
=⇒
(2.10)
def
Indeed, the subspace Vb = span{x`q : q = 1, . . . , d} of V represented
by v is Ad(H)-invariant, and by taking derivatives, it is ad(h)-invariant.
Suppose j ∈ {1, . . . , m} ∩ J (i0 ) and r ∈ {m + 1, . . . , m + n}, r 6= i0 .
Then by definition i0 6= j and Ei0 ,j ∈ Vb . Since Ej,r ∈ h, we have
−Ei0 ,r = ad(Ej,r )Ei0 ,j ∈ Vb .
This implies r ∈ J (i0 ), proving (2.9). The proof of (2.10) is similar.
It follows that for each i0 we have
X
tj ≤ 0 .
j∈J (i0 )
Indeed, if J (i0 ) ⊂ {m+1, . . . , m+n} this is an immediate consequence
of (1.4),and otherwise this follows from (2.9) and (1.4). Since the set
of indices {(iq , jq ) : q = 1, . . . , p} is a proper subset of
{(i, j) : 1 ≤ i 6= j ≤ m + n} ,
the same reasoning shows that
m+n
X
X
tj ≤ −btc
i0 =1 j∈J (i0 )
By a similar argument,
χ(t) =
Pm+n P
p
X
tiq −
q=1
=
m+n
X
i∈I(j0 ) ti
j0 =1
p
X
≥ btc. Therefore
tjq
q=1
X
j0 =1 i∈I(j0 )
ti −
m+n
X
X
tj ≥ 2btc
i0 =1 j∈J (i0 )
which finishes the proof due to (1.8) and (2.8).
In view of Lemma 2.3, Theorem 2.2 can be immediately derived from
the following result:
10
DMITRY KLEINBOCK AND BARAK WEISS
Theorem 2.4. Let H be a connected unipotent subgroup of G, let ν
be a Haar measure on H, and let {gk } be a sequence of diagonalizable
elements in G which normalize H and satisfy
v ∈ V H r {0} =⇒ ρ(gk )v →k→∞ ∞ .
(2.11)
Then for any x ∈ G/Γ, any ψ ∈ L1 (H, ν) with kψk1 = 1 and any
continuous compactly supported function f on G/Γ we have
Z
Z
lim
f (gk hx)ψ(h) dν(h) =
f dµ.
k→∞
H
G/Γ
Moreover, this convergence is uniform as x ranges over compact subsets
in G/Γ.
2.4. The cone lemma revisited. The deduction of Theorem 2.4 from
the results of [SW] is given at the end of this section.
A large class of examples in which condition (2.11) is satisfied is
given by epimorphic subgroups, which we now recall. One says that a
subgroup P of G is epimorphic in G if for any representation ρ : G →
GL(V ) of G one has V P = V G . As an example let H be as in (2.6); it
is well-known that the group P = AH is epimorphic in G; see [We, SW]
and the references therein.
Let a be the Lie algebra of A, let exp : a → A and log : A → a
be the exponential and logarithmic map respectively. We will say that
A0 ⊂ A is a cone if log A0 is a convex cone. A basic general result
about epimorphic groups, from which (2.11) may be deduced, is the
following:
Lemma 2.5 (Cone Lemma). If AH is an epimorphic subgroup of G,
where H is unipotent and A is diagonalizable and normalizes H, then
for any representation ρ : G → GL(V ) there is a nonempty open cone
A+ = A+ (ρ) in A such that for any divergent sequence {gk } ⊂ A+ and
any v ∈ V H r V G we have ρ(gk )v → ∞.
The proof given in [We] does not construct the cone A+ explicitly.
The true meaning of Lemma 2.3 is a precise determination of A+ (ρ),
where ρ is the representation of G on V as in (2.7). Namely, it is
identified with a+ by means of the ‘exponential’ map t 7→ gt .
Proof of Theorem 2.4. We apply the proof of [SW, Thm 1.4] with G =
L, Γ = Λ given by (1.9). Note that in this case Ad is faithful and so
may be omitted from the notation. The assumptions in [SW] regarding
AH and T ++ are needed only to ensure, in the notation of [SW], that
{Ad ai } is C(ρ ⊕ ρ̄)-divergent. Since in our case ρ̄ = ρ, this condition is
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
11
precisely what was proved in Lemma 2.3. We also note in passing that
the assumption that AH is epimorphic is a consequence of (2.11). 3. Theorem 1.5 and quantitative nondivergence
3.1. A sufficient condition. The second goal of this paper is to show
that sets DIε (T ) are null with respect to certain measures µ on Mm,n
other than Lebesgue. We will use Proposition
2.1 to formulate a condi
tion sufficient for having µ DIε (T ) = 0 for fixed ε > 0 and all T ∈ a+
drifting away from walls. Similarly to the set-up of Theorem 1.5, we
will consider measures µ of the form F∗ ν, where ν is a measure on Rd
and F a map from Rd to Mm,n .
Proposition 3.1. Let a measure ν on Rd , an open subset U of Rd , a
map F : U → Mm,n , and 0 < ε, c < 1 be given.
(a) Fix T0 ⊂ a+ and assume that for any ball B ⊂ U there exists
t > 0 with
ν x ∈ B : gt τ̄ F (x) ∈
/ Kε ≤ cν(B)
(3.1)
whenever t ∈ T0 , |tk > t. Then F∗ ν DIε (T ) = 0 for any
unbounded T ⊂ T0 .
(b) Suppose that for any ball B ⊂ U there exists s > 0 with (3.1)
satisfied for any t ∈ a+ with btc ≥ s. Then F∗ ν DIε (T ) = 0
for any T ⊂ a+ drifting away from walls.
Proof. Taking an unbounded T ⊂ T0 , we infer from the assumption of
part (a) that for any ball B ⊂ U and any positive t one has
\
ν
{x ∈ B : gt τ̄ (Y ) ∈
/ Kε } ≤ cν(B) .
t∈T , ktk>t
Therefore, by (2.3), ν {x ∈ B : F (x) ∈ DIε (T )} ≤ cν(B). In view
of a density theorem for Radon measures
on Euclidean spaces [Mat,
−1
Corollary 2.14], this forces F
DIε (T ) to have ν-measure zero. This
establishes (a). The second part is immediate from (a) and the definition (1.8) of drifting away from walls.
3.2. A quantitative nondivergence estimate. During the last decade,
starting from the paper [KM2], a powerful technique has been developed allowing to prove measure estimates as in (3.1) for a certain broad
class of measures ν and maps F .
Let us now introduce the key definitions mentioned in the introduction. If B = B(x, r) is a ball in Rd and c > 0, cB will denote the
ball B(x, cr). A measure ν on Rd is said to be D-Federer on an open
12
DMITRY KLEINBOCK AND BARAK WEISS
U ⊂ Rd if for all balls B centered at supp ν with 3B ⊂ U one has
ν(3B)/ν(B) ≤ D.
If ν is a measure on Rd , B a subset of Rd with ν(B) > 0, and f a
real-valued function on B, we let
def
kf kν,B =
sup
|f (x)| .
x∈B∩supp ν
Given C, α > 0, open U ⊂ Rd and a measure ν on Rd , say that f :
U → R is (C, α)-good on U with respect to ν if for any ball B ⊂ U
centered in supp ν and any ε > 0 one has
α
ε
ν {y ∈ B : |f (y)| < ε} ≤ C
ν(B) .
(3.2)
kf kν,B
We need to introduce some more notation in order to state a theorem
from [KLW]. Let
def
W = the set of proper nonzero rational subspaces of Rk .
From here until the end of this section, we let k · k stand for the Euclidean norm on Rk , induced by the standard inner product h·, ·i, which
we extend from Rk to its exterior algebra. For V ∈ W and g ∈ G, let
def
`V (g) = kg(v1 ∧ · · · ∧ vj )k ,
where {v1 , . . . , vj } is a generating set for Zk ∩ V ; note that `V (g) does
not depend on the choice of {vi }.
Theorem 3.2 ([KLW], Theorem 4.3). Given d, k ∈ N and positive
constants C, D, α, there exists C1 = C1 (d, k, C, α, D) > 0 with the
following property. Suppose a measure ν on Rd is D-Federer on a ball
e centered at supp ν, 0 < ρ ≤ 1, and h is a continuous map B
e→G
B
such that for each V ∈ W,
e with respect to ν,
(i) the function `V ◦h is (C, α)-good on B
and
e
(ii) k`V ◦hkµ,B ≥ ρ, where B = 3−(k−1) B.
Then for any 0 < ε ≤ ρ,
ν x ∈ B : π h(x) ∈
/ Kε ≤ C1 (ε/ρ)α ν(B) .
3.3. Checking (i) and (ii). At this point we restrict ourselves to the
set-up of Theorem 1.5, that is consider measures on Rn ∼
= M1,n of the
form f∗ ν, where ν is a measure on Rd and f = (f1 , . . . , fn ) is a map from
an open U ⊂ Rd with ν(U ) > 0 to Rn . In order to combine Proposition
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
13
3.1 with Theorem 3.2, one need to work with functions `V ◦ht for each
V ∈ W, where
def
ht = gt ◦ τ ◦ f ,
(3.3)
and find conditions sufficient for the validity of (i) and (ii) of Theorem
3.2 for large enough t ∈ T .
The explicit computation that is reproduced below first appeared in
[KM2]. Let e0 , e1 , . . . , en be the standard basis of Rn+1 , and for
I = {i1 , . . . , ij } ⊂ {0, . . . , n},
i1 < i2 < · · · < ij ,
(3.4)
def
let e = e ∧ · · · ∧ eij ; then {eI | #I = j} is an orthonormal basis of
Vj I n+1 i1
(R ). Similarly, it will be convenient to put t = (t0 , t1 , . . . , tn ) ∈
a+ where
)
(
n
X
+
n+1
a = (t0 , t1 , . . . , tn ) ∈ R
: t1 , . . . , tn < 0, t0 = −
ti . (3.5)
i=1
Then one immediately sees that for any I as in (3.4),
eI is an eigenvector for gt with eigenvalue etI ,
(3.6)
def P
where tI = i∈I ti . We remark that in view of (3.5), etI is not less
than 1 whenever for any t ∈ a+ and 0 ∈ I, and, moreover,
etI ≥ ebtc
∀ t ∈ a+ and ∀ I ( {0, . . . , n} containing 0 .
(3.7)
Since the action of τ (y), where y ∈ Rn is a row vector, leaves e0
invariant and sends ei , i > 0, to ei + yi e0 , one can write3
(
eI
if 0 ∈ I
P
τ (y)eI =
(3.8)
eI + i∈I ±yi eI∪{0}r{i} otherwise.
Now take V ∈ W, choose a generating set {v1 , . . . , vj } for Zk ∩ V , and
def
expand
P w = v1 ∧ · · · ∧ vj with respect to the above basis by writing
w = I⊂{0,...,n}, #I=j wI eI . Then one has
!
X
X
X
τ (y)w =
wI eI +
wI +
±wI∪{i}r{0} yi eI .
(3.9)
0∈I
/
0∈I
i∈I
/
In the next lemma we summarize two implications of the above formula:
3The
choice of + or − in (3.8) depends on the parity of the number of elements
of I less than i and is not important for our purposes. See however [Kl3] for a more
precise computation.
14
DMITRY KLEINBOCK AND BARAK WEISS
V
Lemma 3.3.
(a) For any w ∈ (Rn+1 ) and t ∈ Rn+1 , the map
y 7→ gt τ (y)w is affine; in other words, for any I ⊂ {0, . . . , n}
the projection of gt τ (y)w onto eI has the form
hgt τ (y)w, eI i = c0 +
n
X
ci yi
(3.10)
i=1
for someVc0 , c1 , . . . , cn ∈ R.
(b) If w ∈ j (Zn+1 ) r {0}, with 1 ≤ j ≤ n (in particular, if w
represents a nontrivial rational subspace of Rn+1 ) and t ∈ a+ ,
then there exists I such that the absolute value of one of the
coefficients ci in (3.10) is at least ebtc .
Proof. Part (a) is immediate from (3.9). For (b), since the coordinates wI of w are integers and at least one of them is nonzero, one
can conclude that there exists I 3 0 such that either |wI | ≥ 1 or
|wI∪{i}r{0} | ≥ 1 for some i ∈
/ I. Therefore, in view of (3.9), (3.6) and
(3.7), the conculsion of (b) follows for this choice of I.
3.4. Proof of Theorem 1.5. The next theorem generalizes [KM2,
Theorem 5.4]:
Theorem 3.4. For any d, n ∈ N and any C, α, D > 0 there exists
C2 = C2 (d, n, C, α, D) with the following property. Suppose a measure
e centered at supp ν and f : B̃ → Rn
ν on Rd is D-Federer on a ball B
is continuous. Assume that:
(1) any linear combination of 1, f1 , . . . , fn is (C, α)-good on B̃ with
respect to ν;
e
(2) the restrictions of 1, f1 , . . . , fn to B ∩ supp ν, where B = 3−n B,
are linearly independent over R.
Then there exists s > 0 such that for any t ∈ a+ with btc ≥ s and any
ε < 1, one has
ν x ∈ B : gt τ̄ f (x) ∈
/ Kε ≤ C2 εα ν(B) .
Proof. We would like to apply Theorem 3.2 with h = ht as in (3.3).
Take V ∈ W and, as before, represent it by w = v1 ∧ · · · ∧ vj , where
{v1 , . . . , vj } is a generating set for Zk ∩ V . From Lemma 3.3(a) and
assumption (1) above it follows that for any t, each coordinate of ht (·)w
is (C, α)-good on B̃ with respect to ν. Hence the same, with C replaced
α/2
by n+1
C, can be said about kht (·)wk = `V ◦ ht , see [KLW, Lemma
j
4.1]. This verifies condition (i) of Theorem 3.2.
Now observe that assumptionP
(2) implies the existence of δ > 0
(depending on B) such that kc0 + ni=1 ci fi kν,B ≥ δ for any c0 , c1 , . . . , cn
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
15
with max |ci | ≥ 1. Using Lemma 3.3(b), we conclude that
k`V ◦ ht kν,B ≥ δebtc .
So condition (ii) of Theorem 3.2 holds with ρ = 1 whenever btc is at
def
least s = − log δ.
We can now proceed with the
Proof of Theorem 1.5. It suffices to show that for ν-a.e. x there exists
a ball B centered at x such that
ν {x ∈ B : f (x) ∈ DIε (T )} = 0 .
(3.11)
Since (f , ν) is (C, α)-good and nonplanar, for ν-a.e. x one can choose
B centered at x such that (f , ν) is nonplanar on B and (C, α)-good on
3n B ⊂ U . Then, combining Theorem 3.4 with Proposition 3.1(b), one
concludes that (3.11) holds whenever C2 εα < 1.
We remark that the constant C1 from Theorem 3.2, and hence C2
from Theorem 3.4 and ε0 from Theorem 1.5, can be explicitly extimated
in terms of the input data of those theorems, see [KM2, BKM, KT, Kl3].
However we chose not to bother the reader with explicit computations,
the reason being that in the special cases previously considered in the
literature our method produces much weaker estimates. More on that
in the next section.
4. Examples
4.1. Polynomial maps. A model example of functions which are (C, α)good with respect to Lebesgue measure is given by polynomials: it is
shown in [KM2, Lemma 3.2] that any polynomial of degree ` is (C, α)good on R with respect to λ, where α = 1` and C depends only on
`. The same can be said about polynomials in d ≥ 1 variables, with
1
α = d`
and C depending on ` and d. Obviously λ is 3d -Federer on
Rd . Thus, as a corollary from Theorem 1.5, we obtain the existence of
ε1 = ε1 (n, d, `) such that whenever ε < ε1 , (1.10) holds for ν = λ and
any polynomial map f = (f1 , . . . , fn ) of degree ` in d variables such
that 1, f1 , . . . , fn are linearly independent over R. This in particular
applies to f (x) = (x, . . . , xn ), a generalization of the set-up of Theorem
1.2 considered by Baker in [Ba1] for n = 3 and then by Bugeaud for
an arbitrary n. Note that it is proved in [Bu] that (x, . . . , xn ) is almost
surely not in DIε for ε < 1/8. Our method, in comparison, shows that
(x, . . . , xn ) is almost surely not in DIε (T ) for any T drifting away from
2
walls and ε < 1/nn (n + 1)2 2n +n .
16
DMITRY KLEINBOCK AND BARAK WEISS
4.2. Nondegenerate maps. Here is another situation in which (C, α)good functions arise. The following lemma is a strengthening of [KM2,
Lemma 3.3], which can be proved by a verbatim repetition of that
paper’s argument:
0
Lemma 4.1. For any d, ` ∈ N there exists positive Cd,`
with the followd
ing property. Let B be a cube in R (product of intervals of the same
length), and let f ∈ C ` (B) be such that for some positive constants
a1 , . . . , ad and A1 , . . . , Ad one has
ai ≤ |∂i` f (x)| ≤ Ai
∀ x ∈ B, i = 1, . . . , d .
Then
λ {x ∈ B : |f (x)| < ε} ≤
0
Cd,`
max
i
Ai
ai
1/` ε
kf kλ,B
1/d`
λ(B) .
Recall that a map f from U ⊂ Rd to Rn is called `-nondegenerate
at x ∈ U if partial derivatives of f at x up to order ` span Rn , and
`-nondegenerate if it is `-nondegenerate at λ-a.e. x ∈ U . Arguing as in
the proof of [KM2, Proposition 3.4], from the above lemma one deduces
Proposition 4.2. For any d, ` ∈ N there exists positive Cd,` with the
following property. Let n ∈ N and let f = (f1 , . . . , fn ) : U → Rn be
`-nondegenerate at x ∈ U . Then for any C > Cd,` there exists a neighborhood V ⊂ U of x such that any linear combination of 1, f1 , . . . , fn
is (C, 1/d`)-good on V .
This implies that if f : U → Rn is `-nondegenerate, then the pair
(f , λ) is (C, 1/d`)-good for any C > Cd,` . Also, nonplanarity is clearly
an immediate consequence of nondegeneracy. Thus, by Theorem 1.5,
there exists ε2 = ε2 (n, d, `) such that whenever ε < ε2 , (1.10) holds
for ν = λ and any `-nondegenerate f : U → Rn , U ⊂ Rd . This was
previously established in the case d = 1, l = n = 2 in [Ba2], with an
additional assumption that f be C 3 rather than C 2 . Also, M. Dodson,
B. Rynne, and J. Vickers considered C 3 submanifolds of Rn with ‘twodimensional definite curvature almost everywhere’, a condition which
implies 2-nondegeneracy (and requires the dimension of the manifold
to be at least 2). It is proved in [DRV] that almost every point on such
n
a manifold is not in DIε for ε < 2− n+1 .
4.3. Friendly measures. The class of friendly measures was introduced in [KLW], the word ‘friendly’ being an approximate abbreviation of ‘Federer, nonplanar and decaying’. Using the terminology of
the present paper, we can define this class as follows: a measure µ on
Rn is friendly if for µ-a.e. x ∈ Rn there exist a neighborhood U of x
DIRICHLET’S THEOREM AND HOMOGENEOUS FLOWS
17
and D, C, α > 0 such that µ is D-Federer on U , and (Id, µ) is both
(C, α)-good and nonplanar on U . In order to apply Theorem 1.5 we
would like to use somewhat more uniform version: given C, α, D > 0,
define µ to be (D, C, α)-friendly if for µ-a.e. x ∈ Rn there exists a
neighborhood U of x such that µ is D-Federer on U and (Id, µ) is both
(C, α)-good and nonplanar on U .
As discussed in the previous subsections, smooth measures on nondegenerate submanifolds of Rn satisfy the above properties. Furthermore,
the class of friendly measures is rather large, many examples are described in [KLW, KW1, Ur1, Ur2, SU]. A notable class of examples
is given by limit measures of finite irreducible systems of contracting
similarities [KLW, §8] (or, more generally, self-conformal contractions,
[Ur1]) of Rn satisfying the open set condition, as well as their pushforwards by smooth nondegenerate maps [KLW, §7]. We remark that
all the friendly measures considered in the literature, can be shown to
be (D, C, α)-friendly for some D, C, α. Thus, in view of Theorem 1.5,
almost all points with respect to those measures are not in DIε (T ) for
any T drifting away from walls and small enough ε.
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Brandeis University, Waltham MA 02454-9110 [email protected]
Ben Gurion University, Be’er Sheva, Israel 84105 [email protected]