Khintchine's theorem and transference principle for star bodies

0.3 MB·English

by  M. M. Dodson

KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR
STAR BODIES
M. M. DODSON AND S. KRISTENSEN
6
0
0
Abstra
t. Analogues of Khint
hine's Theorem in simultaneous Diophantine approxi-
2
mationintheplaneareprovedwiththe
lassi
alheightrepla
edbyfairlygeneralplanar
n
distan
efun
tionsorequivalentlystarbodies. Khint
hine'stransferen
eprin
ipleisdis-
a
J
ussed for distan
e fun
tions and a dire
t proof for the multipli
ative version is given.
A transferen
e prin
iple is also established for a di(cid:27)erent distan
e fun
tion.
2
1

]
T
1. Introdu
tion
N
S Rn
.
h A star body in Eu
xlid∈eaRnnspa
e is de(cid:28)tn∈ed(0a,s∞an] open set
ontta<intingtxth∈e Sorigin
t and for whi
h given any , there exists a 0 su
h that for 0, and
a t > t tx ∈/ S
m for 0, (see [8℄). To su
h sets, one
an asso
iate a
ontinuous distan
e fun
tion
F : Rn → [0,∞) x ∈ Rn t ≥ 0 F(tx) = tF(x)
[ su
h that for any and any , . The open
S F(x) < 1
1 star body may be expressed as the set of points satisfying . Conversely, we
v
an asso
iate an open star body to any distan
e fun
tion by this relation. Some authors
6

onsider only star bodies symmetri
about the origin, but this restri
tion is not needed
8
2 here and we
onsider the more general
ase. However, we restri
t ourselves prin
ipally
n = 2
1
to the planar
ase ( ), as in higher dimensions the number of
ases to be
onsidered
0
6 proliferates and geometry be
omes very
ompli
ated. R2 |(x ,x )|
0 Examples of planar distan
e fun
tions in
lude the height on given by 1 2
= max{|x |,|x |} ∞
/ 1 2
h , where the asso
iated star body is a square. Another example is the
(x ,x ) 7→ |x ||x |
t 1 2 1 2
a fun
tion given by where the asso
iated star body is bounded by
x = 1/x x = −1/x
m 2 1 2 1
the hyperbola and ipts mirror image . Su
h symmetri
star bodies
v: and their relationship to latti
es have been studied extensively. We refer to [8℄ for an
i ex
ellent introdu
tion.
X
Inthispaper, westudytherelationshipbetweenplanarstarbodiesandmetri
Diophan-
r F : R2 → [0,∞) ψ : N → (0,∞)
a tine approximation. Let be a distan
e fun
tion and let
qψ(q)
be a fun
tion with non-in
reasing. For
onvenien
e we study the set
W(F;ψ) = x ∈ T2 : F(x−p/q) < ψ(q) p ∈ Z2
for some
q ∈ N
(cid:8)
for in(cid:28)nitely many
∞ ∞ (cid:9)
= x ∈ T2 : F(qx−p) < qψ(q) ,
(1)
N=1q=Np Z2
\ [ [∈ (cid:8) (cid:9)
T2 {(x ,x ) ∈ R2: 0 ≤ x ,x < 1} F
1 2 1 2
where denotes the two-torus . When is the height
|·| W(F;ψ) ψ F
, isthesetofsimultaneously -approximablepoints. When isthefun
tion
∞
2000 Mathemati
s Subje
t Classi(cid:28)
ation. 11J83, 11H16.
Key words and phrases. Diophantine approximation; distan
e fun
tions; metri
theory.
1
2 M. M. DODSONANDS. KRISTENSEN
(x ,x ) 7→ |x ||x | W(F;ψ) ψ
1 2 1 2
, is the set of multipli
atively -approximable points. In
p 0 ≤ |p| < q
both of thepse spe
ial
ases, we
an restri
t to those points with without
loss of generality. As we shall see, su
h restri
tions are not natural in the general
ase.
|E| E ⊆ R2
Let denote the Lebesgue measure of . Khint
hine's Theorem (see e.g. [7℄),
qψ(q)
whi
h asserts that when is de
reasing,
0 ∞ q2ψ(q)2 < ∞,
|W(|·| ;ψ)| = if q=1
∞ (1 ∞ q2ψ(q)2 = ∞, (2)
if Pq=1
P
was extended by Gallagher [12℄ who established a general theorem implying (2) and the
following:
W((x ,x ) 7→ |x ||x |;ψ) = 0 if ∞q=1q2ψ(q)2log(q−1ψ(q)−1) < ∞,
(cid:12) 1 2 p 1 2 (cid:12) (1 if P∞q=1q2ψ(q)2log(q−1ψ(q)−1) = ∞. (3)
(cid:12) (cid:12)
(cid:12) (cid:12) P
Evidently both these results are of Khint
hine type, where the measure of a set of points
ψ
-approximable by a distan
e fun
tion is either null or full, a

ording to the
onvergen
e
or divergen
e of an `area' series.
The above results suggest that a general Khint
hine-type theorem might exist for star
bodies. While Gallagher's result implies Khint
hine type theorems for a number of star
bodies, only
onvex star bodies and star bodies of the general shape as the ones in (3)
are
overed. The te
hni
al requirement for Gallagher's result (property P) is that given
(p ,p ) S (0,0) (p ,0)
1 2 1
a point inside the star body , the entire re
tangle with verti
es , ,
(p ,p ) (0,p ) S
1 2 2
and must be a subset of .
It would also seem that a growth
ondition on the size of the unbounded parts of the
star body would in(cid:29)uen
e the breaking point (i.e., when the sum
onverges or diverges).
In an earlier paper [9℄, the (cid:28)rst author found a measure zero result of the above type
under a te
hni
al
overing
ondition. This work was subsequently generalised to systems
of linear forms [10℄. However, the spe
i(cid:28)
star bodies to whi
h these results
an easily
be applied are only minor generalisations of the examples above.
Some re
ent results in Diophantine approximation on manifolds due to Bernik, Klein-
bo
k and Margulis [4, 14℄ suggest that there might be a
onne
tion between distan
e
fun
tions and Diophantine approximation and that a general Khint
hine-type theorem
mightexist forgeneral star bodies. In Se
tion 2 below, we willprove Khint
hine type the-
orems for a large
lass of distan
e fun
tions. The results depend
riti
allyon the intrinsi
arithmeti
properties of the distan
e fun
tions as well as their geometri
properties. We
shall state the results on
e we have the notational apparatus in pla
e.
AtantalisingresultduetoAlievandGruber[1℄isina
ertainsense dualtotheproblem
studied inthis paperand measures theset oflatti
eswhi
h have pointsinsidea given star
S ⊆ Rn
body of in(cid:28)nite volume. They show that given a star body of in(cid:28)nite volume,
n S
almost all latti
es have linearly independent primitive points inside .
W(|·| ;ψ) W((x ,x ) 7→ |x ||x |;ψ)
1 2 1 2
Asso
iated with the sets and are two transfer-
∞
en
e prin
iples, whi
h relate the simultaneous Diophantinpe approximation of the
oordi-
x ∈ Rn q ·x x ∈ Rn q ∈ Zn hxi = x−k ∈
x
nates of to the linear forms , where and . Let
[1/2,1/2)n k ∈ Zn x ∈ Rn
x
, where denotes the symmetrised distan
e from to the nearest
Zn
point in the integer latti
e . The following Transferen
e Prin
iple (see [18℄; a di(cid:27)erent
KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES 3
formulation is in [7, Chapter V, Theorem IV℄) relates the properties of simultaneous Dio-
phantine approximation, asso
iated with the height, and those of the dual inner produ
t
form.
n ∈ N x ∈ (R\Q)n
Khint
hine's Transferen
e Prin
iple. Let and let . The following
two
onditions are equivalent:
ε > 0 q ∈ Zn
(i) For some , there are in(cid:28)nitely many su
h that
|hq ·xi| ≤ |q|−n−ε.
ε > 0 p ∈ Z
′
(ii) For some , there are in(cid:28)nitely many su
h that
kqxk ≤ |q|−(1+ε′)/n.
ε > 0
Note that if
ondition (i) is true for some , it is automati
ally satis(cid:28)ed for all
εˆ< ε
, and similarly for
ondition (ii).
There is an analogous multipli
ativeversion formultipli
ativeDiophantine approxima-
n
tion and the inner produ
t form. Sprindºuk [18, page 69℄ states it without proof (an -th
is root missing from the left hand side of his inequality (9)). Dyson [11℄ dedu
es it from a
more general form of Khint
hine's Transferen
e Prin
iple and [19℄ gives a
ontrapositive
form using a method of A. Baker [2℄.
n ∈ N x = (x ,...,x ) ∈ (R\Q)n
1 n
Theorem 1.1. Let and . The following two
onditions
are equivalent
ε > 0 q ∈ Zn
(i) For some , there are in(cid:28)nitely many su
h that
n −1−ε
|hq·xi| ≤ max(|q |,1) .
i
! (4)
i=1
Y
ε > 0 p ∈ Z
′
(ii) For some , there are in(cid:28)nitely many su
h that
n 1/n
|hpxii| ≤ |p|−(1+ε′)/n.
! (5)
i=1
Y
In Se
tion 3, we shall give a dire
t proof of Theorem 1.1 whi
h also gives a pro-

edure for proving transferen
e prin
iples for other symmetri
distan
e fun
tions than
(x ,...,x ) 7→ ( |x |)1/n
1 n i
. We will also outline an appli
ation to a distan
e fun
tion
not
overed by pQrevious results. In this
ase, the assumption that the star bodies are
symmetri
about the origin is
riti
al.
x,y
We will use the Vinogradov notation. That is, for two real numbers , we will write
x ≪ y C > 0 x ≤ Cy x ≪ y y ≪ x
if there exists a
onstant su
h that . If and , we will
x ≍ y
write .
2. Khint
hine type theorems
We will treat two separate
lasses of distan
e fun
tions, for whi
h the theorems are
of a di(cid:27)erent nature. We make the appropriate de(cid:28)nitions and state the results before
going on to prove the main theorems. In the (cid:28)nal part of this se
tion, we will dis
uss the
remaining distan
e fun
tions and
onje
tures about the
orresponding Khint
hine type
theorems.
4 M. M. DODSONANDS. KRISTENSEN
F : R2 → [0,∞)
2.1. Notation, de(cid:28)nitions and statement of results. Let be a
F
distan
e fun
tion. We de(cid:28)ne the skeleton of to be the set
skel(F) = F 1(0).
−
(6)
It is easy to see, that the skeleton of a distan
e fun
tion
onsists of the origin together
with a (possiblyin(cid:28)nite and possiblyzero) number of half-linesstartingatthe origin. For
ea
h su
h line, the star body has an unbounded
omponent. If there are no su
h lines,
the star body is bounded. Note that Gallagher's property P is only of interest when any
half-lines in the skeleton lie on one of the
oordinate axes.
L F L
Let be a half-line from the skeleton of . We will
all signi(cid:28)
ant if the
omponent
M > 0
around it
arries an in(cid:28)nite amount of mass of the star body, i.e., if for any ,
x ∈ R2 : F(x) < 1 ∩ x ∈ R2 : dist(x,L) < M = ∞.
(cid:12)(cid:8) (cid:9) (cid:8) (cid:9)(cid:12)
Note that the ex(cid:12)isten
e of a signi(cid:28)
ant line in the skeleton of the s(cid:12)tar body immediately
implies that the measure of the star body is in(cid:28)nite. Conversely, if the measure of an
unbounded star body with only (cid:28)nitely many half-lines in its skeleton is in(cid:28)nite, then
the skeleton
ontains a signi(cid:28)
ant line.
F
When the skeleton of
ontains an irrationally sloped signi(cid:28)
ant line, we prove the
following result, generalising a result by the se
ond author [15℄.
F : R2 → [0,∞)
Theorem 2.1. Let be adistan
efun
tion
orrespondingto an unbounded
ψ : N → (0,∞)
star body and let be some fun
tion. Suppose that that at least one of
skel(F)
the (half-)lines in is signi(cid:28)
ant with an irrational slope. Suppose further that for
ε > 0 w(r) = w(r;ε) > 0 F 1([0,ε))
−
ea
h , the width of this unbounded
omponent of
r

ontaining this line is non-in
reasing as the distan
e from the origin in
reases. Then,
x ∈ R2 q ∈ N p ∈ Z2
for almost all and any , there are in(cid:28)nitely many su
h that
F(x−p) < ψ(q),
(7)
when
e
|W(F;ψ)| = 1.
Thus the existen
e of an irrational signi(cid:28)
ant line implies the stronger
on
lusion that
x ∈ R2 ε p ∈ Z2 F(x−p) < ε
foralmostall andforany ,therearein(cid:28)nitelymany forwhi
h .
Note that it is possible for a star body to be unbounded while having no signi(cid:28)
ant lines
in the skeleton. We willreturn to this
ase and others like itlater. While the irrationality
of the slope makes the result plausible, the result is not always true and a monotoni
ity

ondition is
riti
al in the proof. The proof is not straightforward and uses re
ent work
of Bugeaud on a variant of the usual form of Diophantine approximation [6℄.
F
In the
ase when the skeleton of
onsists of (cid:28)nitely many half-lines whose slopes are
rational numbers (or in the degenerate
ase when the skeleton is just the origin), we will
F n
need additional de(cid:28)nitions. Let be su
h a distan
e fun
tion, let be the number of
F s /r
i i
lines in the skeleton of and suppose that the slopes of these are for some integers
s ,r i = 1,...,n {x ∈ R2 : F(x) < ρ}+p
i i
, . We see that if we take the union of the sets
p ∈ Z2 sˆ= lcm(s ,...,s )
1 n
where , we get a pattern whi
h repeats itself with period in the
rˆ= lcm(r ,...,r )
1 n
dire
tion of the (cid:28)rst
oordinate axis and with period in the dire
tion
R F
F
of the se
ond axis. We
all su
h a re
tangle a fundamental re
tangle for .
KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES 5
ε > 0 D ε
F
Let and de(cid:28)ne the fun
tion of by
{x ∈ R2 : F(x) < ε}+p ∩R
p Z2 F
D (ε) = ∈ .
F (cid:12)(cid:12)(cid:16)S |RF| (cid:17) (cid:12)(cid:12) (8)
(cid:12) (cid:12)
When the star body is bounded, the appropriate de(cid:28)nition of the above quantity is
D (ε) = x ∈ R2 : F(x) < ε .
F
(9)
(cid:12)(cid:8) (cid:9)(cid:12)
We
an now state our se
ond Khin(cid:12)t
hine type theorem.(cid:12)
F : R2 → R skel(F)
Theorem 2.2. Let be a distan
e fun
tion su
h that
onsists of
ψ : N → (0,∞)
(cid:28)nitely many lines, ea
h with a rational slope. Let be a fun
tion su
h
D (qψ(q))
F
that is de
reasing. Then
|W(F;ψ)| = 0 whenever ∞q=1DF(qψ(q)) < ∞,
(1 whenever P∞q=1DF(qψ(q)) = ∞.
P
Note that Theorem 2.2
overs (2), (3) and a number of additional
ases, in
luding all
distan
e fun
tions
orresponding to bounded star bodies as well as all the
ases
overed
|R |
F
by Gallagher's result [12℄. Note also, that depends only on the distan
e fun
tion,
and so the series in the statement of Theorem 2.2
ould be repla
ed by volume sums, as
is the
ase in [12℄.
q ∈ N
2.2. The irrational
ase. In this se
tion, we prove Theorem 2.1. Let be (cid:28)xed.
Let
L = {(x,αx): x ∈ [0,∞)}
skel(F) α = tanθ
denote a line through the origin in with irrational slope . Suppose that
α > 1
, as the other
ase may be treated analogously by inter
hanging the axes. Identify
L
ea
h with its
anoni
al proje
tion into the unit square, whi
h is in turn identi(cid:28)ed with
T2 L
the two-torus . Sin
e has irrational slope, it follows from Krone
ker's Theorem (see
T2
e.g. [13, Proposition 1.5.1℄) that ea
h of these geodesi
s is dense in . However, this is
not su(cid:30)
ient to imply Theorem 2.1 for signi(cid:28)
ant lines.
H = {(x,y ): 0 ≤ x < 1} 0 ≤ y <
0 0
Considera(cid:28)xedbutarbitraryhorizontalline ,where
1 T2 H T1
, through the unit square ; will be identi(cid:28)ed with the
ir
le (one dimensional
torus). Consider the geodesi
y
L = ,y : y ∈ [0,∞) .
α
n(cid:16) (cid:17) o
((y +n)/α,y +n) = (y /α,y )+n(1/α,1) L
0 0 0 0
The point on is at distan
e
r = (y +n)/α)2 +(y +n)2 1/2 = (y +n)cosecθ
n 0 0 0
(10)
(cid:0) (cid:1)
from the origin and proje
ts to the point
y +n y +n
0 ,{y +n} = 0 ,y ∈ T2.
0 0
α α
(cid:18)(cid:26) (cid:27) (cid:19) (cid:18)(cid:26) (cid:27) (cid:19)
L H
As the slope of is irrational, the geodesi
interse
ts the horizontal line (cid:28)rst at the
(x ,y ) α > 1
0 0
point , where sin
e ,
y
0
x = x (α) = < 1
0 0
α
6 M. M. DODSONANDS. KRISTENSEN
(Tnx ,y ) n = 1,2,... T: T1 →
0 0
and thereafter at the distin
t points , where and the map
T1 α

an be regarded as a rotation of the
ir
le by the irrational angle and
Tnx := x = {x +nα 1}.
0 n 0 −
L
n
Let be the dire
ted segment whi
h meets
(Tnx ,y ) = (x ,y ) = ({x +nα 1},y )
0 0 n 0 0 − 0
r L ,L ,...
n 0 1
from below. The distan
e along the irrational line segments from the origin
(x ,y ) = ({x +nα 1},y )
n 0 0 − 0
to the point is given by (10).
L
Suppose a signi(cid:28)
ant line. Assume for the sake of simpli
ity that the star body
(x,y)
is approximately symmetri
about signi(cid:28)
ant lines, that is, for any point on a
L (x,y) F 1(ψ(q))
−
signi(cid:28)
ant line , the distan
e from to the upper part of the boundary
w =
±
is
omparable to the distan
e from the lower part. More pre
isely, denote by
w (r,ψ(q)) (x,y) ∈ L r2 = x2+y2
±
the perpendi
ular distan
es of the point , where , from
+ − ∂F 1([0,ψ(q)))
−
the upper ( ) and lower ( ) boundaries of the starbody, so that the
w = w(r,ψ(q)) ψ(q) (x ,y ) w = w+ + w
n 0 −
width of the -neighbourhood at is given by .
w+ = w+((y +n)cosecθ,ψ(q)) w w
Writing n 0 and similarly for n− and n, we have
w = w+ +w ,
n n n−
w+ ≍ w ≍ w
where n n− n.
(y −ρ ,y +ρ ) (y +α 1−ρ ,y +α 1+ρ )
0 0 0 0 1 − 1 1 − 1 H =T1
I I
n n+1
Figure 1. Two
onse
utive intervals and
I H ψ(q)
n
Let be the interval formed by the interse
tion of and the -neighbourhood of
L (x ,y ) |I |
n n 0 n
the segment and its
ontinuation through (see Figure 1). Then the length
I
n
of the interval satis(cid:28)es
w cosecθ ≪ |I | ≪ w cosecθ,
n n n
w ≍ |I | (x ,y )
n n n 0
that is, . Ins
ribe an interval
entred at with radius
σ = Kmin{w+,w }.
n n n−
K
where is a su(cid:30)
iently small positive
onstant
hosen so that
I˜ := x ∈ T1: x− x −nα 1 < σ ⊆ I ,
n 0 − n n
(11)
(cid:8) (cid:13) (cid:8) (cid:9)(cid:13) (cid:9)
(cid:13) (cid:13)
KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES 7
kxk
where denotes the distan
e to nearest integer (this is possible by the assumption of
σ ≍ w |I˜ | ≍ |I |
n n n n
approxqim∈aNte symmetry(oxf,tyhe) ∈staHr body). Then I˜ ⊆aInd . We shonw that
0 n n
given , almost all lie in the interval for in(cid:28)nitely many .
E(y ) x (x,y )
0 0
Considerthe`limsupI˜'set H ofpoints su
h thatthepoint fallsintoin(cid:28)nitely
n
many of the intervals in . Now
I˜ ≍ |I |
n n
Xn (cid:12) (cid:12) Xn
(cid:12) (cid:12)
(cid:12) (cid:12)
and
|I | ≍ σ ≍ w(r ,ψ(q)) ≍ w((y +n)cosecθ,ψ(q)).
n n n
n n n n
X X X X
But
|I | ≍ w(ucosecθ,ψ(q))du ≍ w(v,ψ(q))dv = ∞,
n
n Z Z
X
w(u,ψ(q)) L
sin
e by assumption is non-in
reasing and sin
e the line is signi(cid:28)
ant. Thus
I˜ = ∞.
n
(12)
Xn (cid:12) (cid:12)
E(y ) (cid:12) (cid:12)
0 (cid:12) (cid:12)
We now show that has full measure. By [3℄ it su(cid:30)
es to prove lo
al ubiquity of
x +{n/α} n = 1,2,... λ
0
the points , relative to an appropriate fun
tion . This relies on
the so-
alled Three Distan
es Theorem due to Sós [17℄ and to a
ertain extent follows the
methods used by Bugeaud in [6℄ (see also [16℄ for an approa
h using
ontinued fra
tions).
λ : (0,∞) → (0,∞) (N )
r
Let be a fun
tion de
reasing to zero and let be a stri
tly
R = {z : n ∈ N}
n
in
reasing sequen
e of naturalnumbers. The set of points inthe metri
T1 λ (N ) κ > 0
r
spa
e is said to be lo
ally ubiquitous relative to and if there is a
onstant
I = (c−ρ,c+ρ) ρ r ≥ r (I)
0
su
h that for any interval with su(cid:30)
iently small and any ,
µ I ∩ (z −λ(N ),z +λ(N )) ≥ 2κρ.
n r n r
(cid:18) Nr≤n[<Nr+1 (cid:19)
R
This quantitative version of the lo
al density of the points in is the appropriate spe-

ialisation of the de(cid:28)nition of lo
al ubiquity in [3℄.
R = {{x + nα 1} : n ∈ N} (R,n)
0 −
Lemma 2.3. Let . The system is lo
ally ubiquitous
λ(N) = 3/(N +1) N ≤ N < N
r r r+1
relative to the fun
tion , where for an appropriately
(N )
r

hosen stri
tly in
reasing sequen
e of natural numbers .
Proof. As intheproofof [6, Proposition1℄, we (cid:28)nd thatby theThree Distan
es Theorem,
N ∈ N
for any , the sequen
e of points
{x +α 1},{x +2α 1},...,{x +Nα 1}
0 − 0 − 0 −
T1 N +1
partition the
ir
le into intervals with lengths from a set of
ardinality at most
{x} x
three, where denotes the fra
tional part of . The largest of these distan
es may be
3/(N +1) N
seen to be less than or equal to for in(cid:28)nitely many positive integers . Call
(N ) N ∈ [N ,N ]
r r r+1
this sequen
e . Hen
e, for any , the intervals
entred at the points
{α 1},...,{Nα 1} 3/(N +1)
− − r
ofλlength(N )
over the
ir
le. Therefore the system is lo
al(cid:3)ly
r
ubiquitous relative to and .
8 M. M. DODSONANDS. KRISTENSEN
˜
I
n
Consider the family of intervals . The
entres of these intervals form a ubiquitous
λ
system relative|tI˜o| and it is an easy
onsequeEn
(eyo)f [3, Theorem 2℄ that the divergen
e
of the sum n n implies that the limsup set 0 has full measure. It follows that for
y ∈ [0,1) p ∈ Z2
0
any (cid:28)xed P , for almost there exist in(cid:28)nitely many su
h that
F((x,y )−p) < ψ(q).
0
χ E(y)
E(y)
Let denote the
hara
teristi
fun
tion of . Theorem 2.1 now follows from
Fubini's theorem, sin
e
1 1 1
χ (x,y)dxdy = χ (x,y)dxdy = 1 dy = 1,
E(y) E(y)
ZZ[0,1)2 Z0 Z0 Z0
q ∈ N
when
e a fortiori, for any ,
(x,y) ∈ T2 : F((x,y)−p) < ψ(q) p ∈ Z2 = 1.
for in(cid:28)nitely many
(cid:12)(cid:8) (cid:9)(cid:12)
In the m(cid:12)ore general
ase, i.e., when the star body is not assumed to be(cid:12)approximately
symmetri
about signi(cid:28)
ant lines, the above argument
an be extended by
hanging the
I
n
resonant points in the ubiquitous system to the
entres of the intervals . These do not
have to be as regularly spa
ed as the resonant points
onsidered in the spe
ial
ase, but
λ
this di(cid:30)
ulty
an be resolved by adding an additional term to the ubiquity fun
tion in
Lemma 2.3. In this
ase, the ubiquity fun
tion should be
λ(N) = 3/(N +1)+max{w+ ,w }.
r Nr N−r
max{w+ ,w }
Ifne
essary, wemaytakeasmallerstarbodyinsidetheoriginaloneforwhi
h Nr N−r
isnotdominatingthemainterm, butforwhi
hthe
orrespondinglineremainssigni(cid:28)
ant.
The details are left to the interested reader.
F 1([0,ε))
−
Note that if the unbounded starbody has no signi(cid:28)
ant lines (so that has
q ∈ N (x,y) ∈ T2
(cid:28)nite volume), then given , the set of points for whi
h
F((x,y)−p) < ψ(q)
p ∈ Z2
for in(cid:28)nitely many is null by the Borel(cid:21)Cantelli lemma. The question of the
(x,y)
measure of the set of for whi
h
p
F (x,y)− < ψ(q)
q
(cid:18) (cid:19)
p/q
holds for in(cid:28)nitely many is open.
2.3. The rational
ase. For
ompleteness, we (cid:28)rst deal with the degenerate
ase when
F
the skeleton
onsists solely of the origin,
orresponding to being a gauge fun
tion [8℄.
S ⊆ R2
Lemma 2.4. Let be a bounded star body with
orresponding distan
e fun
tion
F
. Then
|W(F;ψ)| = 0 if ∞q=1DF(qψ(q)) < ∞,
(1 if P∞q=1DF(qψ(q)) = ∞.
P
KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES 9
c,C > 0
Proof. By the (cid:28)nal
orollary of [8, Se
tion IV.2℄, there are
onstants su
h that
x ∈ R2
for any ,
c|x| ≤ F(x) ≤ C|x| ,
(13)
∞ ∞
when
e ψ ψ
W |·| ; ⊆ W(F;ψ) ⊆ W |·| ; .
C c
∞ ∞
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:3)
Applying (2) immediately gives the result.
We now suppose that the skeleton of the star body
ontains a half-line and prove the
q ∈ N ε > 0
two
ases of Theorem 2.2 separately. First, for any and any , we de(cid:28)ne the
resonant neighbourhood
B (F,ε) = x ∈ T2 : F(x−p/q) < ε .
q
(14)
p Z2
[∈ (cid:8) (cid:9)
D (ε)
F
The measure of the resonant neighbourhood is
losely related to . Note (cid:28)rst that
sˆrˆ
1R ∩B (F,ε) = D (qε),
q F q q2 F (15)
(cid:12) (cid:12)
(cid:12) (cid:12) q
as is seen by s
aling the set t(cid:12)o be estimated b(cid:12)y and using homogeneity of the distan
e
fun
tion.
B (F,ε) R2 1R
To obtainthe measure of q , we tile by disjoint translates of q F. Inside ea
h
1R ∩B (F,ε)
of these disjoint sets, we (cid:28)nd a translated
opy of q F q , and the union of these
B (F,ε) B (F,ε)
q q
sets
over . Hen
e, to estimate the measure of , it su(cid:30)
es to
ount the
1R
maximal (resp. minimal) number of disjoint translates of q F that
an (cid:28)t inside (resp.
are needed to
over) the unit square, and multiply the result by the estimate from (15).
In this way, we get
q q sˆrˆ q q sˆ rˆ
D (qε) ≤ |B (F,ε)| ≤ +1 +1 D (qε),
rˆ sˆ q2 F q sˆ rˆ q2 F
h ih i (cid:16) (cid:17)(cid:16) (cid:17)
when
e
|B (F,ε)| ≍ D (qε).
q F
(16)
Applying the Borel(cid:21)Cantelli Lemma [5, Lemma 3.14℄ to the resonant neighbourhoods
|W(F;ψ)|
with the measure estimate (16) yields a
ondition for to be zero. Indeed,
whenever
∞
D (qψ(q)) < ∞,
F
(17)
q=1
X
W(F;ψ)
must be null. We have shown:
ψ : Z → (0,∞) F : R2 → R
Lemma 2.5. Let be some de
reasing fun
tion and let be
skel(F) D (ε)
F
a distan
e fun
tion su
h that the lines in all have rational slopes. Let be
de(cid:28)ned as above. Suppose that
∞
D (qψ(q)) < ∞.
F
q=1
X
|W(F;ψ)| = 0
Then .
10 M. M. DODSONANDS. KRISTENSEN
We now aim to obtain the
orresponding divergen
e result, i.e., when the series (17)
W(F;ψ)
diverges, we expe
t the measure of to be full. Suppose (cid:28)rst that the
entral
F
part of the star body
arries the bulk of the mass, i.e., there is a distan
e fun
tion
su
h that
F e
(i) determines a bounded star body.
F(x) ≤ F(x) x F
(ii) for all (or equivalently, the star body de(cid:28)ned by is a subset of the
e F
one de(cid:28)ned by by [8, Corollary, p. 107℄).
D (ε) ≍eD (ε) ε > 0 e
(iii) F F for small enough.
W(F;ψ) ⊆ W(F;ψ)
In this
ease, , so the
on
lusion of Theorem 2.2 is ensured by Lemma
2.4. We have shown
e
F skel(F)
Lemma 2.6. Let be a distan
e fun
tion and suppose that
onsists of lines with
F
rational slopes. Suppose further that there is a distan
e fun
tion satisfying (i)(cid:21)(iii)
above. Then Theorem 2.2 holds for this distan
e fun
tion.
e
Suppose now that the
entral part of the star body does not
arry the bulk of the mass,
so that there is a
usp whi
h in
ludes an in(cid:28)nite amount of the mass. We will
onstru
t
W(F;ψ)
a subset of , whi
h is better behaved but whi
h still has full measure. In brief,
we will trun
ate the star body in su
h a way that only the unbounded
omponent whi
h

arries most of the mass will remain. Subsequently, we will take the limsup set of these
F
ε∗
trun
ated sets, for whi
h we
an
al
ulate the measure. For
onvenien
e we denote by
{dist(x,L) < w(ε)}∩{F(x) < ε}
the distan
e fun
tion asso
iated with the star body .
F
Lemma 2.7. Let be a distan
e fun
tion as in the statement of Theorem 2.2. Then
w : (0,∞) → (0,∞) w(ε) ε
there exists a fun
tion with tending to zero as tends to zero
L ∈ skel(F)
together with a (half-)line su
h that
DF∗(1) ≍ DF(ε),
ε (18)
L
and so that the interse
tion between any line perpendi
ular to and the star body asso-
F
ε∗

iated to is a line segment.
Remark: We have to
hoose an entire family of distan
e fun
tions rather than just a
single one. This is to preserve the s
aling properties of the original star body. This takes
us away from the problem of approximation with respe
t to distan
e fun
tions, but as
we shall see, only the shape of the sets studied are of importan
e for the remainder of
the proof.
skel(F)
Proof. We
hoose the line in about whi
h a signi(cid:28)
ant proportion of the mass is
skel(F)

on
entrated. Su
h a line must exist, sin
e there are only (cid:28)nitely many lines in .
By trun
ating the star body about this line, it is now straightforward to
hoose the
w
fun
tion in su
h a way that (18) holds. To ensure that the se
ond property holds, we
w skel(F)

hoose as small as possible. For at least one of the lines in the , we will have
both properties satis(cid:28)ed. Indeed, otherwise mFost of the mass would be
on
entrated i(cid:3)n
the
ental part of the star body asso
iated to .
F ,F : R2 → R skel(F ) = skel(F ) = L
1′ 2′ 1′ 2′
Lemma 2.8. Let be distan
e fun
tions with , a
δ ,δ > 0 v ∈ R2 L v
1 2 ⊥
(half-)line. Let . Let be a unit ve
tor in the dire
tion of and let be