Queen Mary ~ Westfield College. London, England ... award NSF 9623045. The second author is a Royal Society University Research Fellow. .... In any case, given our current state of knowledge the result of Cassels and. Swinnerton-Dyer ...
Acta Math., 185 (2000), 287-306 @ 2000 by Institut Mittag-Lefller. All rights reserved
On a problem in simultaneous Diophantine approximation: Littlewood's conjecture by ANDREW D. POLLINGTON
and
SANJU L. VELANI
Queen Mary ~ Westfield College London, England, U.K.
Brigham Young University Provo, UT, U.S.A.
Dedicated to Bill Parry on his 65th birthday
1. I n t r o d u c t i o n 1.1. B a c k g r o u n d : e l e m e n t a r y n u m b e r t h e o r y Before stating the problem we recall a few fundamental results from the theory of Diophantine approximation. Given a real number x we use the standard notation [[xI[ to denote the distance of x to the nearest integer, and throughout I will denote the unit interval [0, 1]. The classical result of Dirichlet states: DIRICHLET'S THEOREM (1842). For any aEI:=[O, 1], there exist infinitely many
q E N such that
IIqII< q-1 A consequence of Hurwitz's theorem is that the right-hand side of the above inequality cannot be improved by an arbitrary positive constant s. More precisely, for s < l / x / 5 there exist real numbers a E I for which the inequality Ilqall ~ 0 so that
IIq~I[> c ( a ) q - '
for all q G N } .
We now briefly describe the beautiful connection between B a d and the theory of continued fractions. Let (~=-[al,a2,a3, ...] represent the regular continued fraction expansion of a, and as usual let Pn/qn : : [al, a2, a3, ..., an] denote its nth convergent. It is This work was supported in part by the National Science Foundation under award NSF 9623045. The second author is a Royal Society University Research Fellow.
288
A.D. POLLINGTON AND S.L. VELANI
easy to verify that 1
1
a n + l + 2 [[q~aN for any q 1,
(2)
where K ( a ) is an absolute constant; t h a t is, the denominators of the convergents form a
lacunary sequence and qn IIq~c~ll~ 1;
(3)
that is, the left-hand side is bounded from above and below by constants independent of n. We will make use of all these elementary facts later. For further details and proofs see [4], [11], [12]. It is clear from the above discussion t h a t B a d is uncountable. A simple consequence of a fundamental result due to Khintehine is that B a d is a set of zero Lebesgue measure. KHINTCHINE'S THEOREM (1924). Let ~b be a real positive function and let W(~b):= {xE I : Ilqxll 0 ~ (F) >1)~/c. In particular, dim F ~>s.
then
Pro@ If {Ci} is a 0-cover of F with 00, the last part, dimF~>s, follows immediately from the definition of Hausdorff dimension.
[]
We will also work with the Fourier transform of a measure. The Fourier transform of a measure m supported on a subset X of R is defined by fit(t) := I x exp(27ritx) dm(x),
t ER.
The decay rate of the transform is related to lower bounds for the dimension of X. We will not require the relationship, but for completeness we mention that if for some r]>0 then dimX>~min{1,2r]}.
Ifit(t)l
8 , 4 N l o g 2 ~dim G ( a ) ~>dim G n ( a ) ~> 1 On letting N - + o o we conclude that dim G ( a ) = l ,
and this completes the proof of the
theorem assuming of course that # ((1N (a)) > 0 - - this we now prove.
3.1. P r o o f o f t h e c l a i m t t ( G N ( a ) ) > 0 Let ~ be a real positive decreasing function such that
q~(q)-+O as q-+ec. For q C N let
q--1
Eq(r : : [0,r
U
B(p/q,r
11,
p:l
B(c, r) is an interval centered at c with radius r. We now estimate the Kaufman Eq(~). The following result shows that if the interval width determined by the function ~ is sufficiently large then the Kaufman measure of Eq(r is essentially equal to the Lebesgue measure of Eq(r that is, to the total length of the disjoint intervals defining Eq( ~ ). where
measure of
LEMMA 1.
For N)3, #( Eq(r )) = 2q~(q) + O(q-'#2).
Proof. We fix q and for the sake of clarity put & = r characteristic function defined by X~(x)=
1
if Ilxll ~5,
Let x~:R--+R be the
LITTLEWOOD'S CONJECTURE
295
and let X~,e: R--+R be the continuous approximation of Xa given by
Xs+~(x) =
1
if [[x[[ ~ 6+e,
where 0 < r The function X6,~ + is normally referred to as the 'upper smoothed' characteristic function, and obviously X~,~(x)>~Xa(x) for all x in R. Clearly, X+~,~is a periodic function with period 1. Next consider the function W 5,r + defined by
w~,~(x) := (~ sp/~(x)) *x~i,~(z),
(5)
where as usual * denotes convolution and 5a is the Dirac delta-function. It is easily verified that q--1
w~,Ax) = F_, x~,~(x-;/q), p=O and so it follows that
#(Eq(~p)) < S01W~,~(x) d#(x). We now proceed to evaluate the integral by considering the Fourier series expansion of W +5,~. For k c Z , let ~.~(k), and W~,~(k) denote the kth Fourier coefficient of X~,~ and W 6,~ + respectively. A straightforward calculation shows that
2~,~(k) =
{
26+a
if k = 0 ,
cos(27ckS)-cos(2~rk(cf+r
if k # 0 .
(~)
27r2k2r
Since W 6,6 + is defined via convolution, we have that
q-1
~4,~(k) := Z G/~(k) ~,Ak) p=O
Trivially, 6p/q(k)=exp(-2~ikp/q).
Thus it follows from (6) that for k # 0 ,
q(cos( 27ckS) -cos( 27ck( 6 + c ) ) )
(r) if qJ(k,
0 and for k=0, A
W+~ (0) = 25q+qr
(8)
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A.D.
POLLINGTON
AND
S.L. VELANI
A
Clearly E k e Z
IW~+,a(k)l~ s W[,e(x ) d#(x) ~ 25q-q~
#(Eq(r
3qt+~e"
This together with (9) implies that
,u(Eq(~b)) = 25q+O(qe+c/q 1+~?C).
(10)
The lemma now follows on setting e=l/q 1+'7/2.
[]
We now put q=q~ (the denominator of the n t h convergent of c~) and r
1/q~ log q~. We will often make use of the fact that KI I
This follows from the fact that the sequence qn is
are constants.
lacunary and that it satisfies the recurrence qn = an q~-i + q,~-2 for n ~>2. Also let qn E n : = Eq,~(~(qn)) = [.J U ( p / q n , l / q n 10g qn) A I . p=0
By definition, GN(C~) is precisely the set of real numbers in FN which lie in infinitely m a n y of the sets En; t h a t is, oG
:=FNCl ~]
GN(C~) = F N n l i m s u p E n n-+oo
U En.
rn=l n=m
Recall that our aim is to show that # ( G N ( a ) ) > 0 . Note that since p is supported on FN we trivially have that # ( G N ( a ) ) = #(lim sup En). n---~or
By L e m m a 1,
1
1
p ( E n ) ~ log qn -- n
(12)
since K~ I such that for Q sufficiently large, E
#(EmNE~) ~ lira sup (~-~Q-1 m(A~)) 2 Q ~-~ O-*~ ~,~,~=I m(AmnA~)" In our situation, the proposition together with the divergent sum (13) and the quasiindependence on average result implies that ~(GN(~)) ~> i/C > O. This completes the proof of the claim assuming of course the quasi-independence on average result--this we now prove.
3.2. P r o o f o f L e m m a 2: q u a s i - i n d e p e n d e n c e on a v e r a g e In view of (12) and (13), it is sufficient to prove that for Q sufficiently large,
E
(14)
#(EmnEn)~1 > (mn)2+e(st) -1. Thus the second sum in (25) when summed over
l
