Simultaneous Diophantine approximation - logarithmic improvements

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by  Alexander Gorodnik

SIMULTANEOUS DIOPHANTINE APPROXIMATION — LOGARITHMIC
IMPROVEMENTS
ALEXANDERGORODNIKANDPANKAJVISHE
Abstract. This paper is devoted to the study of a problem of Cassels in multiplicative
6 Diophantine approximation which involves minimising values of a product of affine linear
1 forms computed at integral points. It was previously known that values of this product
0 become arbitrary close to zero, and we establish that, in fact, they approximate zero with
2
an explicit rate. Our approach is based on investigating quantitative density of orbits of

n higher-rank abelian groups.
a
J

4 1. Introduction
1
Let hui denote the distance of the real number u to the nearest integer. The sequence

] hqui with q ∈ N reflects how well u is approximated by rational numbers. In particular, it is
T
well-known that for every Q ≥ 1 one can findq ≤ Q such that hqui≤ 1/Q, butthere is a large
N
set of numbers u satisfying hqui ≥ c(u)/q for all q’s with some c(u) > 0. The long-standing
.
h Littlewoodconjectureconcernssimultaneous approximationofapairofrealnumbersu,v ∈ R.
t
a It asserts that
m
(1) liminfqhquihqvi = 0
[
q→∞

1 holds for all u,v ∈ R. This paper deals with the inhomogeneous version of this problem,
v
5 namely, whether the following relation
2
5 (2) liminf|q|hqu−αihqv−βi= 0
3 |q|→∞
0 holds for u,v,α,β ∈ R. In this setting Cassels asked (see [4, p. 307]) whether there exists
.
1 a pair (u,v) for which the property (2) holds for all real numbers α,β. This question was
0
answered affirmatively by Shapira in [16] who showed that this is true for almost all pairs
6
1 (u,v). He also gave an explicit example of a family of algebraic numbers (u,v) satisfying this
: property, and showed that it fails if u and v are rationally dependent.
v
i Itisnaturaltoask whethertheLittlewood conjecture(1)andits inhomogeneousversion (2)
X
admit quantitative improvements. It follows from the results of Gallagher [7] that for almost
ar every (u,v) ∈ R2,
(3) liminf (logq)2qhquihqvi= 0.
q→∞
Peck [14] showed if 1,u,v form a basis of a real cubic field, then
(4) liminf (logq)qhquihqvi < ∞.
q→∞
Pollington and Velani [15] proved that (1) holds with an additional logq factor for a large set
of pairs (u,v), and Badziahin and Velani [2] conjectured that (4) holds for all real numbers u
and v.
Unlike in the homogeneous setting, literature on quantitative results in the inhomogeneous
setting has been lacking. An old argument of Cassels readily implies that for almost all
(u,v,α,β) ∈ R4,
liminf(logq)2qhqu−αihqv−βi = 0
q→∞
1
2 ALEXANDERGORODNIKANDPANKAJVISHE
(see, for instance, [9, Theorem 3.3]). The case with α = 0 was investigated by Haynes, Jensen
and Kristensen in [10]. They proved that for all badly aproximable u, and v contained in a
set of badly approximable numbers of full Hausdorff dimension depending on u,
liminf(logq)1/2−ǫqhquihqv−βi = 0 with any ǫ > 0
q→∞
holds for all β. Setting α = 0 allowed in [10] to use tools developed in [15], but it seems
unlikely that this approach could be applied when α is non-zero.
Apart from these results, no other quantitative improvements of the inhomogeneous prop-
erty (2) are known to us. The aim of this paper is to establish the first quantitative improve-
ment of (2) with arbitrary α,β. In contrast with the existing analytical methods, dynamical
ideas employed in this paper enable us to successfully deal with general α,β at a cost of a
weaker logarithmic saving. The following theorem is a quantitative refinement of one of the
main results from [16].
Theorem 1. There exists δ > 0 such that for almost all (u,v) ∈ R2,
liminf (log |q|)δ|q|hqu−αihqv−βi = 0
(5)
|q|→∞
holds for all α,β ∈R. Here log denotes the s-th iterate of the function max(1,log|x|).
(s)
We note that our method, in principle, could also allow establishing this result for spe-
cific pairs (u,v) provided that corresponding orbits satisfy a certain quantitative recurrence
property.
In a subsequent paper [8], we also extend Theorem 1 to the p-adic setting motivated by the
p-adic version of the Littlewood conjecture proposed by de Mathan and Teuli´e [5].
The setting of Theorem 1 can be considered as a particular case of a general problem of
multiplicative Diophantineapproximationforaffinelattices (alsocalled grids)intheEuclidean
space Rd. A grid in Rd is a subset of the form
Zx +···+Zx +w,
1 d
where x ,...,x ∈ Rd are linearly independent and w ∈ Rd. To formulate this problem
1 d
explicitly, we set N(v) := v v ···v for a vector v = t(v ,...,v ) in Rd.
1 2 d 1 d
Definition 1.1. Let Λ be a grid in Rd and h :R+ → (0,1) a function such that h(x) → ∞ as
x → ∞.
(i) We say that Λ is multiplicatively approximable if 0 is a non-trivial accumulation point
of a sequence N(v ) with v ∈ Λ.
n n
(ii) We say that Λ is h-multiplicatively approximable if there exists a sequence v ∈ Λ such
n
that v → ∞ and 0< |N(v )| < h(kv k)−1.
n n n
Wenotethatthisprovidesanaturalgeneralisationofproperty(2). Indeed,foru,v,α,β ∈ R,
we consider the grid
(5) Λ(u,v,α,β) := {t(x,xu−y−α,xv−z−β) : x,y,z ∈Z}.
It is easy to check that if the grid Λ(u,v,α,β) is multiplicatively approximable, then (2)
holds. Moreover, assuming that the function h is non-decreasing, if the grid Λ(u,v,α,β) is
h-multiplicatively approximable, then
liminfh(c |q|−c )|q|hqu−αihqv−βi ≤ 1
1 2
|q|→∞
for some c ,c > 0.
1 2
It was also proved in [16] that for almost every lattice Λ in Rd, the grid Λ+v is multi-
plicatively approximable for all v ∈ Rd. Here we establish a quantitative refinement of this
result.
Theorem 2. There exists δ > 0 such that for almost every lattice ∆ in Rd, every grid ∆+w,
w ∈ Rd, is h-multiplicatively approximable with h(x) = (log x)δ.
(5)
SIMULTANEOUS DIOPHANTINE APPROXIMATION — LOGARITHMIC IMPROVEMENTS 3
The paper is organised as follows. In the following section we set up required notation and
give a dynamical reformulation of the problem, which reduces our investigation to the study
of a quantitative recurrence property for orbits of a higher-rank abelian group A acting on
the space of grids in the Euclidean space. However, it is not easy to establish this recurrence
propertydirectly, soinSection3, wefirstinvestigate quantitative recurrenceinasmallerspace
— the space of lattices. In particular, it would be crucial in the proof to establish recurrence
to neighbourhoods of lattices with compact A-orbits. In Section 4, we discuss properties of
compact orbits and relevant density results. Finally, in Section 5 we give a proof of the main
theorems performing local analysis in a neighbourhood of a grid whose corresponding lattice
has compact A-orbit.
1.1. Acknowledgements. TheauthorswouldliketothankS.Velaniforsuggestingtheprob-
lemandforhisencouragement duringtheworkontheproject. Thefirstauthorwas supported
by ERC grant 239606, and the second author was supported by EPSRC programme grant
EP/J018260/1.
2. Preliminaries
In this section we introduce some basic notation regarding dynamics on the space of grids
in Rd and give a dynamical reformulation of the above Diophantine approximation problem.
We also introduce a collection of roots subgroup that provides a convenient system of local
coordinates.
2.1. Space of grids. Let G denote the group of unimodular affine transformations of Rd.
Let us set G := SL(d,R) and V := Rd. Then G ≃ V ⋊G . We also set Γ := SL(d,Z) and
0 0 0
Γ := Zd ⋊Γ . Then Γ is lattice in G , and Γ is lattice in G. The space X := G /Γ can
0 0 0 0 0
be identified with the space of unimodular lattices in Rd, and the space Y := G/Γ can be
identified with the space of affine unimodular lattices, which are also called unimodular grids.
For x ∈ X we denote by ∆ the corresponding lattice in Rd, and for y ∈ Y, we denote by Λ
x y
the corresponding grid. We denote by π : Y → X the natural factor map. We observe that
Λ = ∆ +w for some w ∈ V. Moreover, w can chosen to be uniformly bounded when π(y)
y π(y)
varies over bounded subsets of X.
2.2. Dynamical reformulation of the multiplicatively approximable property. We
show that the multiplicatively approximable property can bereformulated in terms of dynam-
ics of the group
A := {a = diag(a ,...,a ): a > 0}
1 d i
acting on the space Y. More specifically, we show that the grid Λ is h-multiplicatively
y
approximable if the orbit Ay visits certain shrinking subsets W(ϑ,ε) of Y. Given ε,ϑ > 0, we
introduce the following non-empty open subsets of Y
W(ϑ,ε):= {y ∈ Y : ∃v ∈ Λ such that kvk < ϑ and 0< |N(v)| < ε}.
y
We also denote by k·k the maximum norm on Mat(d,R), and for a subset S of Mat(d,R),
we set
S(T) := {s ∈ S : ksk < T}.
Proposition 3. Let h be a nondecreasing function such that h(x) → ∞ as x → ∞. Suppose
that for y ∈ Y,
(WR) ∃T → ∞ : A(T )y∩W(T ,h(Td)−1) 6= ∅.
n n n n
Then the grid Λ is h-multiplicatively appoximable.
y
Proof. It follows from our assumption that there exist sequences a(n) ∈ A(T ) and v(n) ∈ Λ
n y
such that
(n) (n)
|a v |< T for all i,
i i n
4 ALEXANDERGORODNIKANDPANKAJVISHE
and
|N(a(n)v(n))| = |N(v(n))| ∈ (0,h(T )−1).
n
This, in particular, implies that 0 6= N(v(n)) → 0, so that v(n) → ∞. We deduce from the
first inequality that
−1
|v(n)|< a(n) T = a(n) T ≤ Td.
i i n  j  n n
(cid:16) (cid:17) Yj6=i
 
Hence, kv(n)k≤ Td, and since h is nondecreasing, we conclude that
n
0< N(v(n))< h(kv(n)k)−1.
This proves that the grid Λ is h-multiplicatively appoximable. (cid:3)
y
Proposition 3 reduces study of the problem of multiplicative approximation to analysing
property (WR) — the quantitative recurrence property of A-orbits with respect to the sets
W(ϑ,ε) in Y.
2.3. Root subgroups. The crucial ingredient in understanding dynamics of the A-action on
the spaces X and Y are the root subgroups, which we now introduce. The adjoint action of A
on the Lie algebra of G is diagonalisable, and we denote by Φ(G) the set of roots A which is
the set of non-trivial eigencharacters of A appearing in this action. For each α ∈ Φ(G), there
is a one-parameter root subgroup Uα = {uα(t)}t∈R ⊂ G such that
au (t)a−1 = u (α(a)t) for a ∈A and t ∈ R.
α α
More explicitly, the set of roots consists of
α (a) = a a−1 for 1 ≤ i6= j ≤ d and β (a) = a for 1 ≤ i≤ d.
ij i j i i
The corresponding root subgroups are the groups of affine transformations defined by
u (t)u = u+tu e and v (t)u = u+te for u∈ Rd,
ij j i i i
where e ,...,e denotes the standard basis of Rd. We denote the set of roots of the first type
1 d
by Φ(G ) and the set of roots of the second type by Φ(V). With a suitable ordering, the
0
product maps
A× R → G : (a,t : α ∈ Φ(G )) 7→ a u (t ) ,
0 α 0 α α
 
α∈YΦ(G0) α∈YΦ(G0)
 
R → V : (t : α ∈ Φ(G )) 7→ u (t ),
α 0 α α
α∈Φ(V) α∈Φ(V)
Y Y
and
A× R → G : (a,t :α ∈ Φ(G)) → u (t ) a u (t )
α α α α α
   
α∈YΦ(G) α∈YΦ(V) α∈YΦ(G0)
   
are diffeomorphisms in neighbourhoods of the origins. We set
U (ε) := {a ∈ A : ka−ek < ε}· {u (t ) :|t |< ε},
G0 α α α
α∈YΦ(G0)
(6) U (ε) := {u (t ) :|t |< ε},
V α α α
α∈Φ(V)
Y
U (ε) := U (ε)U (ε).
G V G0
SIMULTANEOUS DIOPHANTINE APPROXIMATION — LOGARITHMIC IMPROVEMENTS 5
Then U (ε), U (ε), and U (ε) define neighbourhoods of identity in the groups G , V, and G
G0 V G 0
respectively. We also consider the neighbourhoods of identity
O (ε) := {g ∈G : kg−ek < ε},
G0 0
(7) O (ε) := {v ∈ V : kvk < ε},
V
O (ε) := {(v,g) ∈ G : kvk < ε, kg−ek < ε}.
G
It is easy to check that there exists c > 0 such that for every ε ∈ (0,1),
0
(8) U (ε) ⊂ O (c ε), U (ε) ⊂ O (c ε), U (ε) ⊂ O (c ε).
G0 G0 0 V V 0 G G 0
While establishing quantitative recurrence of A-orbits to the sets W(ϑ,ε) is the crux of the
proof of our main results, it turns out that analogous recurrence property is easy to verify for
the root subgroups. In fact, as an intermediate step in the proof, we will have to establish
recurrence to smaller sets which are defined as
W(ϑ,ε ,ε ) := {y ∈ Y : ∃v ∈ Λ such that kvk <ϑ and ε < |N(v)| < ε }
1 2 y 1 2
for ϑ > 0 and 0 < ε < ε .
1 2
Lemma 4. Let α ∈ Φ(G). For every ε ,ε ∈ (0,1), ε < ε , and y ∈ Y, there exist positive
1 2 1 2
ϑ = O (1), positive t = O (1), and negative t = O (1) such that
π(y) + π(y) − π(y)
u (t )y ∈ W(ϑ,ε ,ε ) and u (t )y ∈ W(ϑ,ε ,ε ).
α + 1 2 α − 1 2
Proof. We first note that the grid Λ can be written as Λ = ∆ +w, where w belongs to
y y π(y)
a fixed fundamental domain for the lattice ∆ . In particular, kwk = O (1).
π(y) π(y)
Let us show that v (t)y ∈ W(ϑ,ε ,ε ) for some positive ϑ = O (1) and some positive
i 1 2 π(y)
t = O (1). Using that kwk = O (1) and adding a suitable vector from the lattice ∆ ,
π(y) π(y) π(x)
one can show that there exists a vector z ∈ Λ = w+∆ such that
y π(y)
kzk = O (1), z < 0, |z | ≥ 1 for all k.
π(y) i k
Indeed, since ∆ is a lattice, there exists s ∈ ∆ satisfying s < 0 and s 6= 0 for all k.
π(y) π(y) i k
Then we can choose z of the form z = w+ℓs with a suitable ℓ ∈ N. We have to choose t so
that the inequalities
ε < |N(v (t)z)| < ε
1 i 2
hold. SinceN(v (t)z) = N(z)+tN (z)whereN (z) := z ,theseinequalitiesareequivalent
i i i j6=i i
to
ε |N (z)|−1 < |z +t|< εQ|N (z)|−1.
1 i i 2 i
Hence, we can take t from the interval (ε |N (z)|−1 −z ,ε |N (z)|−1 −z ). Due to our choice
1 i i 2 i i
of z, we have t > 0 and t = O (1). Also, kv (t)zk = O (1). Hence, it follows that
π(y) i π(y)
v (t)y ∈ W(ϑ,ε ,ε ) with some ϑ = O (1) as required. Similarly, one can also show that
i 1 2 π(y)
there exists negative t satisfying v (t)y ∈ W(ϑ,ε ,ε ).
i 1 2
The proof that u (t)y ∈ W(ϑ,ε ,ε ) for some positive ϑ = O (1) and some positive
ij 1 2 π(y)
t = O (1) follows similar lines. Since kwk = O (1), we can add to w a vector from the
π(y) π(y)
lattice ∆ to show existence of z ∈ Λ = w+∆ satisfying
π(x) y π(x)
kzk = O (1), z > 0, z < 0, |z | ≥ 1 for all k.
π(y) i j k
Since N(u (t)z) = N(z)+tz N (z), the inequalities
ij j i
ε < |N(u (t)z)| < ε
1 ij 2
are equivalent to
ε |N (z)|−1|z |−1 < |z z−1+t| < ε |N (z)|−1|z |−1,
1 i j i j 2 i j
so that we can take t from the interval (ε |N (z)|−1|z |−1 −z z−1,ε |N (z)|−1|z |−1 −z z−1).
1 i j i j 2 i j i j
Then t > 0 and t = O (1). Also, it is clear that ku (t)zk = O (1). Hence, it follows that
π(y) ij π(y)
u (t)y ∈ W(ϑ,ε ,ε ) with some ϑ = O (1). The argument with negative t is similar. (cid:3)
ij 1 2 π(y)
6 ALEXANDERGORODNIKANDPANKAJVISHE
3. Quantitative recurrence estimates
QuantitativerecurrenceplaysanimportantroleinthetheoryofDiophantineapproximation.
In particular, this connection was realised in Sullivan’s work [17] and its subsequent general-
ization [12] by Kleinbock and Margulis. While these papers deal with recurrence to shrinking
cuspidalneighbourhoods,wehavetoinvestigate visitsofA-orbitstoshrinkingneighbourhoods
ofspecificpointsinsidethespaceX. Theideaofourapproach,whichusesexponentialmixing,
is similar to [12], but it will be essential to establish recurrence to neighbourhoods of partic-
ular shape with respect to the root coordinate system introduced in Section 2.3. Namely, we
consider neighbourhoods of x ∈ X defined by U (x) := U (ε)x, where U (ε) is defined in
ε G0 G0
(6).
The main goal of this section is to prove the following proposition.
Proposition 5. Let x ∈ X and a be a non-trivial one-parameter subgroup of A. Then there
0 t
exists a constant β > 0, such that for almost every x ∈ X and every T > T (x),
0
atx ∈ UT−β(x0)\UT−β/2(x0) for some t ∈ [0,T].
We denote by µ the normalised invariant measure on the space X and consider a family of
averaging operators
1 T
A : L2(X) → L2(X) :f 7→ f(a x)dt.
T t
T
Z0
We begin by proving an L2-estimate for the operators A .
T
Lemma 6. There exists α > 0 such that for every T ≥ 1 and f ∈ C∞(X),
c
A (f)− fdµ ≪ T−αS(f),
T
(cid:13) ZX (cid:13)2
(cid:13) (cid:13)
where S(f) denotes a suitable(cid:13)Sobolev norm. (cid:13)
(cid:13) (cid:13)
Proof. We recall the exponential mixing property (see, for instance, [11, Sec. 3]): there exists
δ > 0 such that for every f ,f ∈ C∞(X),
1 2 c
(9) f (a x)f (x)dµ(x) = f dµ f dµ +O e−δ|t|S(f )S(f ) .
1 t 2 1 2 1 2
ZX (cid:18)ZX (cid:19)(cid:18)ZX (cid:19) (cid:16) (cid:17)
This property will be used to establish the required L2-bound. Without loss of generality, we
can assume that fdµ = 0. Then using (9), we deduce that for every M > 0,
X
R
kA (f)k2 = T−2 f(a x)f(a x)dµ(x)dtds
T 2 t s
Z(t,s)∈[0,T]2ZX
= T−2 f(a x)f(x)dµ(x)dtds
t−s
Z(t,s)∈[0,T]2ZX
= T−2 +
Z(t,s)∈[0,T]2:|t−s|<M Z(t,s)∈[0,T]2:|t−s|>M!
≪ T−2(TMkfk2+e−δMT2S(f)2).
2
Now the lemma follows by choosing a suitable value of M. (cid:3)
We are now ready to apply a standard Borel-Cantelli type argument to prove Proposition
5.
Proof of Proposition 5. Let β ∈ (0,1), which value will be specified later, and
ΩT := {x ∈ X : atx ∈/ UT−β/2β(x0)\UT−β/2(x0) for all 0 ≤ t ≤ T}.
SIMULTANEOUS DIOPHANTINE APPROXIMATION — LOGARITHMIC IMPROVEMENTS 7
We recall from the previous section that the neighbourhoods U (ε) are ε-cubes with respect
G0
to a suitable smooth coordinate system, so that we can choose a non-negative compactly
supported function f such that
T
supp(fT) ⊂ UT−β/2β(x0)\UT−β/2(x0), fT dµ = 1, S(fT)≪ Tcβ
ZX
with some fixed c > 0, determined by the Sobolev norm. We observe that for x ∈ Ω ,
T
A (f )(x) = 0, so that
T T
2
A (f )− f dµ dµ = |Ω |.
T T T T
ZΩT (cid:12) ZX (cid:12)
On the other hand, by Lemma 6(cid:12), (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2 2
A (f )− f dµ dµ ≤ A (f )− f dµ ≪ (T−αS(f ))2 ≪ T2cβ−2α.
T T T T T T T
ZΩT (cid:12) ZX (cid:12) (cid:13) ZX (cid:13)2
(cid:12) (cid:12) (cid:13) (cid:13)
We pick (cid:12)β ∈ (0,1) sufficient(cid:12)ly small(cid:13)to make the last ex(cid:13)ponent negative. Then the above
(cid:12) (cid:12) (cid:13) (cid:13)
estimates imply that
|Ω | ≪ T−ǫ
T
with some ǫ > 0. Hence, it follows from the Borel-Cantelli lemma that the limsup of the sets
Ω has measure zero. This means that for almost every x ∈ X, we have x ∈/ Ω for all
2k 2k
k ≥ k (x), i.e., for all sufficiently large k, there exists t ∈ [0,2k] such that
0
atx ∈ U2−kβ/2β(x0)\U2−kβ/2(x0).
Finally, given general T ≥ 1, we choose k so that 2k ≤ T < 2k+1. Then [0,2k]⊂ [0,T]. Hence,
for all sufficiently large T, there exists t ∈ [0,T] such that
atx ∈ U2−kβ/2β(x0)\U2−kβ/2(x0) ⊂ UT−β(x0)\UT−β/2(x0).
This completes the proof. (cid:3)
Proposition 5 is sufficient for the proof of Theorem 2, but for the proof of Theorem 1 we
need a more refined recurrence property. We consider the one-parameter subgroup
a := diag(e−(d−1)t,et,...,et),
t
and denote by U the expanding horospherical subgroup of G for a defined by
0 t
(10) U := {g ∈ G : a−1ga → e as t → ∞}.
0 t t
We note that U ≃ Rd−1 and the group U generated by the root subgroups U ,...,U . We
21 d1
prove a recurrence result for orbits staring from points in Ux ⊂ X.
Proposition 7. Let x ,x ∈X. Then there exists a constant β > 0, such that for almost every
0
u∈ U and every T > T (u),
0
atux ∈ UT−β(x0)\UT−β/2(x0) for some t ∈[0,T].
Proof. We note that it will be sufficient to prove Proposition 7 for almost all u contained an
open neighbourhood U of identity in U. Our first goal is to prove an analogue of Lemma 6
0
for averages along U x.
0
We introduce a complementary to U subgroup
W := {g ∈ G : a ga−1 is bounded as t → ∞}.
0 t t
The product map W × U → G is a diffeomorphism in a neighbourhood of identity. We
0
fix a right-invariant Riemannian metric on G which also defines a metric on X = G /Γ .
0 0 0
Let W denote the open σ-neighbourhood of identity in W. We assume that σ and U are
σ 0
sufficiently small, so that the product map W ×U → G is a diffeomorphism onto its image,
σ 0 0
and the projection map g 7→ gx, g ∈ W U , is one-to-one. Let X := W U x ⊂ X. We
σ 0 σ σ 0
note that the invariant measure on W U ⊂ G is the image under the product map of a left
σ 0 0
8 ALEXANDERGORODNIKANDPANKAJVISHE
invariant measure on W and a right invariant measure on U . After suitable normalisation,
σ 0
this measure projects to the measure µ on X . This implies that for every f ∈ C∞(X),
σ c
(11) A (f)(wux)− f dµ = A (f)− fdµ
T T
(cid:13) ZX (cid:13)L2(Wσ×U0) (cid:13) ZX (cid:13)L2(Xσ)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) ≤ (cid:13)A (f)− fdµ(cid:13) ≪ T−αS(f),
T
(cid:13) ZX (cid:13)L2(X)
(cid:13) (cid:13)
where in the last estimate we used Lemma 6. (cid:13) (cid:13)
(cid:13) (cid:13)
We observe that for every wux ∈ W U x and every t > 0,
σ 0
d(a wux,a ux) ≤ d(a wa−1,e) ≪ σ.
t t t t
Hence, it follows from the Sobolev embedding theorem that for a suitable Sobolev norm S,
|f(a wux)−f(a ux)| ≪ σS(f), f ∈ C∞(X).
t t c
This also implies that |A (f)(wux)−A (f)(ux)| ≪ σS(f), and
T T
(12) kA (f)(wux)−A (f)(ux)k ≪ σ|W |1/2S(f).
T T L2(Wσ×U0) σ
Combining (11) and (12), we deduce that
A (f)(ux)− fdµ ≪ (T−α+σ|W |1/2)S(f).
T σ
(cid:13) ZX (cid:13)L2(Wσ×U0)
(cid:13) (cid:13)
Hence, (cid:13) (cid:13)
(cid:13) (cid:13)
A (f)(ux)− fdµ = |W |−1/2 A (f)(ux)− fdµ
T σ T
(cid:13) ZX (cid:13)L2(U0) (cid:13) ZX (cid:13)L2(Wσ×U0)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) ≪ (|W |−1/(cid:13)2T−α+σ)S(f) (cid:13)
(cid:13) (cid:13) σ (cid:13) (cid:13)
≪ (σ−dim(W)/2T−α+σ)S(f),
andtakingσ = T−ǫ forsufficientlysmallǫ > 0,weconcludethatforallT ≥1andf ∈C∞(X),
c
A (f)(ux)− f dµ ≪ T−α′S(f)
T
(cid:13) ZX (cid:13)L2(U0)
(cid:13) (cid:13)
with some α′ > 0. (cid:13) (cid:13)
(cid:13) (cid:13)
Finally, we note that the last estimate is a complete analogue of Lemma 6 for averages
along U x. Now we can apply exactly the same argument as in the proof of Proposition 5 to
0
conclude that almost every u∈ U satisfies the claim of the proposition. (cid:3)
0
4. Compact A-orbits
OurargumentisbasedonstudyingdistributionofA-orbitsinaneighbourhoodofacompact
A-orbit. ThisideagoesbacktothepapersofFurstenberg[6]andBerend[3],andinthecontext
of Cartan actions it was developed by Lindenstrauss, Weiss [13] and Shapira [16]. It would be
sufficient for our purposes to know that there exists x ∈ X with a compact A-orbit. In fact,
0
it is known that every order in a totally real number field gives rise to a compact A-orbit (see,
for instance, [13, Sec.6] for details).
From now on we fix x ∈ X such that Ax is compact. Let B := Stab (x ). It is a discrete
0 0 A 0
cocompact subgroup of A. The group B acts on the fiber π−1(x ) which can be naturally
0
identified with the torus Rd/∆ , where ∆ denotes the lattice corresponding to x . Every
x0 x0 0
y ∈ π−1(x ) corresponds to a grid Λ = ∆ +v with v ∈ V. We say that y = π−1(x ) is
0 y x0 0
q-rational if qv ∈ ∆ . We note that B preserves the set of q-rational elements which has
x0
cardinality qd. Hence, if y ∈ π−1(x ) is q-rational, then the subgroup B := Stab (y) has
0 1 A
finiteindexinB,namely |B :B | ≤ qd. Inparticular, thisimplies thefollowingapproximation
1
property.
SIMULTANEOUS DIOPHANTINE APPROXIMATION — LOGARITHMIC IMPROVEMENTS 9
Lemma 8. There exists c > 0 such that for every a ∈A, one can choose b ∈B satisfying
1
kab−1k≤ exp(cqd).
Our argument involves study of dynamics of the action for the groups B and B in a
1
neighbourhood of the fiber π−1(x ). The crucial part will be played by two quantitative
0
density results that we now state. Thefirst result (Theorem 9), which was proved by Z. Wang
[18], establishes quantitative density in the fibers, and the second result (Proposition 10),
which is deduced from the Baker Theorem, will be used to prove density along orbits of root
subgroups.
We say that y ∈ π−1(x ) is Diophantine of exponent k if Λ = ∆ +v and for some c> 0,
0 y x0
(13) |qv−z| ≥ cq−k+1 for every q ≥ 2 and z ∈ ∆ .
x0
The following theorem allows to establish quantitative density in fibers π−1(x ) of the space
0
Y under a Diophantine condition.
Theorem 9 (Z. Wang [18]). There exist Q ,σ > 0 and c = c(x ) > 0 such that for every
0 0
y ∈ π−1(x ) satisfying (13) and Q ≥ Q , the set B(Qk+2)y is (log Q)−σ-dense in the torus
0 0 (3)
π−1(x ).
0
Wenotethatthisresultisstatedin[18](see[18,Theorem10]) forthestandardtorusRd/Zd
and balls defined by the Mahler measure, but it straightforward to extend it to our setting.
On the other hand, if the point y in the fiber is close to a q-rational point, we will analyse
action of the group B on orbits of the root subgroups and use the following proposition.
1
Proposition 10. There exists η > 1 such that given α ∈ Φ(G) and a subgroup B of B of
1
exponent q, for every M ≥ 1 and t > 0, there exists a ∈ B such that
1
|α(a)−t|≪ qtM−1 and logkak ≪ |logt|Mη+1.
We note that η is precisely the exponent appearing in the Baker estimate (16).
In the proof of Proposition 10 we use the following lemma.
Lemma11. LetS be amultiplicative subgroup of R+ generated bymultiplicatively independent
algebraic numbers λ and λ . Then there exists η > 1 such that for every M ≥ 1 and t > 0,
1 2
there exists s = λℓ1λℓ2 ∈ S satisfying
1 2
|s−t| ≪ tM−1 and |ℓ |,|ℓ |≪ |logt|Mη+1.
1 2
Moreover, if S is an exponent q subgroup of S, then there exists s = λℓ1λℓ2 ∈ S satisfying
1 1 2 1
|s−t|≪ qtM−1 and |ℓ |,|ℓ |≪ |logt|Mη+1.
1 2
Proof. We set a = logλ and a = logλ . By Minkowski’s theorem, for every M ≥ 1, there
1 1 2 2
exists (n ,n ) ∈ Z2\{(0,0)} such that
1 2
(14) |n a +n a |≤ M−1 and |n |,|n | ≪ M.
1 1 2 2 1 2
We set a := n a +n a . We note that a 6= 0 because λ and λ are assumed to be multi-
1 1 2 2 1 2
plicatively independent. It is clear that we can arrange a > 0. Let b := ⌈M−1⌉a. Then
a
M−1
(15) M−1 ≤ b < +1 a ≤ 2M−1.
a
(cid:18) (cid:19)
It follows from the Baker Theorem (see, for instance, [1, Ch. 3]) that there exists η > 1 such
that for all (m ,m ) ∈ Z2\{(0,0)},
1 2
(16) |m a +m a | ≥ max(|m |,|m |)−η.
1 1 2 2 1 2
Hence, we deduce from (14) and (16) that ⌈M−1⌉ ≪ Mη−1, so that b = ℓ a + ℓ a with
a 1 1 2 2
|ℓ |,|ℓ | ≪ Mη. It follows from (15) that the set {ib : |i| ≤ LM} forms a 2M−1-net of the
1 2
interval [−L,L]. Hence, for every t > 0, there exists d= iℓ a +iℓ a such that
1 1 2 2
|d−logt|≪ M−1 and |iℓ |,|iℓ |≪ |logt|Mη+1.
1 2
10 ALEXANDERGORODNIKANDPANKAJVISHE
This implies that |ed−t|≪ tM−1, as required.
q
Toprovethesecondpartofthelemma,weapplytheaboveargumenttotheelements λ and
1
λq that belongto the subgroupS . It follows from (14)that there exists (n ,n ) ∈ Z2\{(0,0)}
2 1 1 2
such that
|n a +n a |≤ qM−1 and |n |,|n | ≪ M.
1 1 2 2 1 2
We set a := n a +n a and b := ⌈qM−1⌉a. Then it follows from (16) that a ≥ qM−η. We
1 1 2 2 a
proceed exactly as in the previous paragraph to prove the second part of the lemma. (cid:3)
Proof of Proposition 10. We write x = g Γ for some g ∈ G . Then B = A∩g Γ g−1. It
0 0 0 0 0 0 0 0
follows that entries of elements in B are eigenvalues of matrices from Γ = SL(d,Z). Hence,
0
entries of elements in B are algebraic numbers. In particular, the group α(B) consists of
algebraic numbers. We apply Lemma 11 to this group. It was proven in [16, Cor. 3.3] that
α(B) is dense in R+ for every α ∈ Φ(G). Since B is finitely generated, this implies that α(B)
must contain two multiplicatively independent elements. Now Proposition 10 follows directly
from Lemma 11. (cid:3)
5. Proof of the main theorems
Theproofofthemaintheoremswillusethedynamicalreformulationofthe‘multiplicatively
approximable’ property stated in Proposition 3. More explicitly, we will establish that for
points y in the space of grids Y, their orbits A(R)y visit the shrinkingsets W(ϑ,(log R)−ζ),
(5)
provided that R is sufficiently large. As the first step, we use the results from Section 3
to deduce that the projected orbits A(R)x, with x = π(y), in the space of lattices X visit
shrinking neighbourhoods of any given point x in X. We apply this observation when the
0
point x has compact A-orbit. This will allow to analyse behaviour of A-orbits locally in a
0
neighbourhood of the fiber π−1(x ). The crucial step of the proof is the following proposition:
0
Proposition 12. Let x ∈ X such that Ax is compact and x ∈ X. We assume that for fixed
0 0
ν,β > 0 and all sufficiently large T,
(17) ∃a0 ∈ A : ka0k ≤ eνT and a0x ∈UT−β(x0)\UT−β/2(x0).
Then there exist ϑ,ζ > 0 such that for any y ∈ π−1(x) and all sufficiently large R,
(18) A(R)y∩W(ϑ,(log R)−ζ)6= ∅.
(5)
We begin by investigating how the recurrence property in Proposition 12 changes under
small perturbations of the base point y. It will be convenient to consider the family of neigh-
bourhoods of y in Y defined by O (y) := O (ε)y, where O (ε) is defined in (7).
ε G G
Lemma 13. Let 0 < ε < 1, ε1/2 ≤ ε < ε , ϑ ≥ 1, and a ∈ A(ε−1/(2d)). Then for every
1 2
y ∈ Y, if
(19) ay ∈ W(ϑ,ε ,ε ),
1 2
then for all y′ ∈ O (y),
ε
ay′ ∈ W(3ϑ,c ε ,c ε )
1 1 2 2
with some c ,c > 0, depending only on ϑ.
1 2
Proof. Since y′ ∈ O (y), we can write y′ = hy with h ∈ O (ε). The element h can be written
ε G
as h = (v,g) with v ∈ V satisfying kvk < ε and g ∈ G satisfying kg − ek < ε. Then
0
ay′ = (aha−1)ay where aha−1 = (av,aga−1). We observe that for x ∈ Mat(d,R), we have
kaxa−1k ≤ kak·ka−1k·kxk ≤ kakd·kxk,
so that since a ∈ A(ε−1/(2d)), we deduce that
kaga−1 −ek ≤ ε−1/2kg−ek < ε1/2.
Also kavk ≤ kakkvk < ε1/2.