On Kleinbock's Diophantine result

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by  Nikolay G. Moshchevitin

On Kleinbo
k's Diophantine result
1
by Ni
ky Mosh
hevitin.
Abstra
t. We give an elementary proof of a re
ent metri
al Diophantine result by D. Kleinbo
k related
to badly approximable ve
tors in a(cid:30)ne subspa
es.
Rd (x ,...,x )
1 d
R1.d+1De(cid:28)nitions and notation. Let be a(Exu
,lxid,e.a..n,xsp)a
e wixth∈thRed
oorxd∈inaRtde+s1 ,
0 1 d
let be a Eu
lidean spa
e with the
oordinates . For or we denote
|x|
9 by its sup-norm:
0 |x| = max |xj| |x| = max |xj|.
0 16j6d or 06j6d
2 A ∈ Rd A ∈ Rd+1

n Consider an a(cid:30)ne subspa
e and de(cid:28)ne the a(cid:30)ne subspa
e in the following way:
u
A = {x = (1,x ,...,x ) : (x ,...,x ) ∈ A}.
J 1 d 1 d

8
B B ⊂ A
] Let be another a(cid:30)ne subspa
e su
h that . De(cid:28)ne
T
B = {x = (1,x ,...,x ) : (x ,...,x ) ∈ B}, B ⊂ A.
N 1 d 1 d
.
h
t Put
a a = dimA = dimA, b = dimB = dimB.
m
[
We de(cid:28)ne linear subspa
es A B
= spanA, = spanB
1
v Rd+1 A B
1
as the smallest linear subspa
es in
ontaining or respe
tively. So
4
A B
5 dim = a+1, dim = b+1.
1
.
6 ψ(T), T > 1 T → +∞
0 Let be a positive valued fun
tion de
reasing to zero as . We de(cid:28)ne an
B ψ
9
a(cid:30)ne subspa
e to be -badly approximable if
0
:
v 1
inf inf |x−y| > 0.
Xi x∈Zd+1\{0} ψ(|x|) y∈B ! (1)
r
a
This de(cid:28)nition is very
onvenient for our exposition.
B
HerBe w=e{wwo}uld like to note that in the
ase whenwt=he(aw(cid:30),n.e...s,uwbs)p∈a
Red has zero dimension (and
1 d
hen
e
onsists of just one nonzero ve
tor ) the de(cid:28)nition (1) gives
1
inf inf |x−tw∗| > 0
x∈Zd+1\{0} ψ(|x|) t∈R (2)
(cid:18) (cid:19)
w∗ = (1,w ,...,w )
1 d
with . Inequality (2) is equivalent to the
ondition that there exists a positive
γ = γ(w)

onstant su
h that
max ||w x|| > γ ·ψ(|x|), ∀x ∈ Z\{0}
j
16j6d (3)
|| · ||
(here denotes the distan
e to the nearest integer). The
ondition (3) is a usual
ondition of
ψ
-badly simultaneously approximable ve
tor. This is an explanation our de(cid:28)nition (1).
1
supported by the grant RFBR (cid:157) 09-01-00371
1
ψ(T),ϕ(T)
In the sequel we
onsider two positive de
reasing fun
tions under the
onditions
ϕ(T) 6 ψ(T), lim ψ(T) = 0.
T→+∞
R > 1
For these fun
tions and de(cid:28)ne
a−b a a
T ϕ(RT) ϕ(R(T +1))
(R) (R)
µ = , λ = − .
T ψ(RT) T T T +1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
2. The result. Now we are ready to formulate the main result of this paper.
R > 1 B ⊂ A 0 6 b = dimB < a =
Theorem 1. Let . Suppose that are a(cid:30)ne subspa
es and
dimA 6 d B ψ ϕ(T) 6 ψ(T)
. Let be a -badly approximable a(cid:30)ne subspa
e. Suppose that and the
series +∞
(R) (R)
µ λ
T T (4)
T=1
X
w ∈ A |w| 6 R

onverges. Then almost all (in the sense of Lebesgue measure) ve
tors su
h that are
ϕ
-badly approximable ve
tors.
b = 0 ψ(T) = T−1/d ψ w
We
onsider a spe
ial
ase and . Then -badly approximable ve
tors are
known as simultaneously badly approximable ve
tors (see [1℄, Chapter 2). Take
ϕ(T) = ψ(T)·(logT)−∆.
In the
ase 1
∆ >
a
the series (4)
onverges. So we have the followAin⊂g Rd
Corollary1. Considerana(cid:30)nesubspa
e . Supposethat there existsabadly approximable
w ∈ A ∆ > 1 A 1
ve
tor . Suppose that a. Then almost all ve
tors from the subspa
e are T1/d(logT)∆-badly
approximable ve
tors.
b = 1 ψ(T) = T−1/d
Here we should note that in the
ase and su
h a result was obtained re
ently
by Dmitry Kleinbo
k (see [2℄, Theorem 4.2) by means of theory of dynami
s on homogeneous spa
es.
R > 1
3. Proof of Theorem 1. Given it is enough to prove that almost all ve
tors
w = (1,w ,...,w ) ∈ A |w | 6 R ϕ
1 d j
su
h that are -badly approximable in the sense of the de(cid:28)nition
≪,≍
(2). In the sequel we do not take
are on the
onstants. All
onstants in the symbols may
d A,B R
depend on ,Csu⊂bsRpad+
e1s and x. ∈ Rd+1 |x|C x C
For a set and a point we de(cid:28)ne the distan
e from to by
|x|C = inf |x−y|.
C
y∈
Consider the set
Ω = {z ∈ Rd+1 : 0 6 z 6 T, max |z | 6 RT, |z|B 6 γ ·ψ(RT)}
T 0 j
16j6d
γ > 0 B ψ
where is a lower bound for the in(cid:28)mum in (1). As is a -badly approximable subspa
e we
see that Ω ∩Zd+1 = {0}
T
1/2
(we use the de(cid:28)nition (1)). Now we observe that any translation of the -dilated set
1
·Ω +c, c ∈ Rd+1
T
2 (5)
2
x,y

onsists of not more than one integer point. Indeed if two di(cid:27)erent integer points belong to the
0 6= x−y ∈ Ω
T
same set of the form (5) then . This is not possible.
Consider the set
Π = {z ∈ Rd+1 : 0 6 z 6 T, max |z | 6 RT, |z|A 6 ϕ(RT)}.
T 0 j
16j6d
ϕ(T) 6 ψ(T) ν ≪ µ(R)
As we see that this set
an be
overed by not more than T T di(cid:27)erent sets of the
Π
T
form (5). So we dedu
e a
on
lusion about an upper bound for the number of integer points in :
# Π ∩Zd+1 ≪ µ(R).
T T (6)
T > 1 (cid:0) (cid:1)
For an integer we
onsider the set
Z = {z = (z ,z ,...,z ) ∈ Zd+1 : z = T, max |z | 6 RT, |z|A 6 ϕ(RT)}
T 0 1 d 0 j
16j6d
of the
ardinality ζ(R) = #Z , 0 6 ζ(R) ≪ Ta.
T T T (7)
ρ > 0 z ∈ Rd+1
For and put U (z) = {y ∈ Rd+1 : |z−y| 6 ρ}.
ρ
Consider the set U U
= (z)
T ϕ(RT)
z
[
z
where the union is taken over all integer points su
h that
z = T, max |z | 6 RT, U (z)∩A 6= ∅.
0 j ϕ(RT)
16j6d
Clearly U U
= (z)
T ϕ(RT)
z[∈ZT
Consider the
one G = {x ∈ Rd+1 : x = t·y, t ∈ R,y ∈ U }
T T
and the proje
tion G
U = A∩ .
T T
Consider the series +∞
mes U
a T
(8)
T=1
X
mes a
a
where stands for the -dimensionalLebesgue measure. By the Borel-Cantellilemma arguments
Theorem 1 follows froUm the
onvergen
e of the series (8()R.)
ζ ϕ(RT)
Note that the set T is a union of not more than T balls (in sup-norm) of the radius .
(R)
U ⊂ A ζ ϕ(RT)/T
Hen
e the set T
an be
overed by not more than T balls of the radius . So in
order to prove the
onvergen
e of the series (8) one
an establish the
onvergen
e of the series
+∞ a
ϕ(RT)
(R)
ζ · .
T T (9)
T=1 (cid:18) (cid:19)
X
3
It follows from (6) that
T
(R) (R)
ζ ≪ µ .
j T
j=1
X
Now the
onvergen
e of the series (9) follows from the
onvergen
e of the series (4) by partial
summation as from (7) we see that
a
ϕ(R(T +1))
ζ(R) · ≪ (ϕ(R(T +1)))a → 0, T → +∞.
T T +1
(cid:18) (cid:19)
Theorem 1 is proved.
Referen
es
[1℄ W.M. S
hmidt, Diophantine Approximations, Le
t. Not. Math., 785 (1980)
[2℄ D. Kleinbo
k, Extremal supspa
es and their submanifolds, GAFA 13:2 (2003), 437 - 466.
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