The Duffin-Schaeffer Conjecture with extra divergence II

0.13 MB·English

by  Victor Beresnevich

THE DUFFIN-SCHAEFFER CONJECTURE
WITH EXTRA DIVERGENCE II
VICTOR BERESNEVICH, GLYN HARMAN,
ALAN HAYNES AND SANJU VELANI
2
1
0
Abstract. In[6]the authorssetouta programmetoprovethe Duffin-Schaeffer Conjec-
2
ture for measuresarbitrarilycloseto Lebesgue measure. Inthis paper we takea new step
n inthisdirection. Givenanonnegativefunctionψ :N→R,letW(ψ)denotethesetofreal
a
J numbers x such that |nx−a| < ψ(n) for infinitely many reduced rationals a/n (n > 0).
Our mainresult is that W(ψ) is of full Lebesgue measure if there exists a c>0 suchthat
5
ϕ(n)ψ(n)

] = ∞.
T nexp(c(loglogn)(logloglogn))
n≥16
N X
.
h
t Keywords: Diophantineapproximation,Metricnumbertheory,Duffin-Schaefferconjecture
a
m Mathematics Subject Classification 2000: 11J83, 11K55, 11K60
[

1
v
0
1 1. Introduction
2
1
. We use the following notation: p denotes a prime number, ϕ(n) is the Euler phi function,
1
0 λ denotes Lebesgue measure on R/Z, and f ≪ g means that the absolute value of f is
2 bounded above by a constant times the absolute value of g.
1
: Let ψ : N → R be a non-negative arithmetical function and for each positive integer n
v
i define En ⊆ R/Z by
X
n
r a−ψ(n) a+ψ(n)
a E := , .
n
n n
a=1 (cid:18) (cid:19)
gcd([a,n)=1
Victor Beresnevich: University of York, Heslington, York YO10 5DD, United Kingdom.
E-mail: [email protected] Tel. [phone].
Glyn Harman: Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom.
E-mail: [email protected].
Alan Haynes: Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom.
E-mail: [email protected].
Sanju Velani: University of York, Heslington, York YO10 5DD, United Kingdom.
E-mail: [email protected].
1
2 V.BERESNEVICH, G. HARMAN,A. HAYNES,AND S.VELANI
Denote the collection of points x ∈ R/Z which fall in infinitely many of the sets E by
n
W(ψ). In other words,
∞
W(ψ) := limsupE := E .
n n
n→∞
m=1 n≥m
\ [
The question we address is:
Question 1. Let ψ be any non-negative arithmetical function. Under what circumstances
is it true that λ(W(ψ)) = 1?
It is very easy to give a necessary condition for this to happen, namely the divergence
of the series:
∞
ϕ(n)
(1) ψ(n).
n
n=1
X
This follows from the Borel-Cantelli Lemma, since
ϕ(n)
λ(E ) ≤ 2 ψ(n)
n
n
and so the convergence of (1) implies that λ(W(ψ)) = 0. It is a central open problem in
metric number theory to show that the divergence of (1) is actually sufficient to conclude
that λ(W(ψ)) = 1.
Conjecture(Duffin-Schaeffer1941). We have thatλ(W(ψ)) = 1 if andonlyif (1) diverges.
There are several significant partial results towards this conjecture, most notably those
due to Khintchine, Duffin & Schaeffer, Erd¨os, Vaaler, and Pollington & Vaughan [7, 2, 3,
10, 8]. The proofs of these results and others are all given in [5, Chps 2 & 3]. Recently
Pollington and two of this paper’s authors [6] have considered the effect on the problem of
assuming “extra divergence”. They have posed the following question.
Question 2. For what functions f does the divergence of
∞
ψ(n)
(2) f ϕ(n)
n
n=1 (cid:18) (cid:19)
X
guarantee that λ(W(ψ)) = 1?
In view of the Mass Transference Principle [1] this question is equivalent to investigating
the Duffin-Schaeffer Conjecture for (Hausdorff) measures “arbitrarily” close to Lebesgue
measure – see [6, §5] for the details. Regarding Question 2 itself, the following result is
established in [6].
DUFFIN-SCHAEFFER WITH EXTRA DIVERGENCE II 3
Theorem HPV. Let ψ be any non-negative arithmetical function and define the function
f by
0 if x = 0,
f(x) = xexp logx if 0 < x < 1,
 log(−logx)

1 (cid:16) (cid:17) if x ≥ 1.
Then the divergence of (2) is sufficient to conclude that λ(W(ψ)) = 1.

The authors of [6] set as an explicit unsolved problem the task of replacing f(x) above
by f(x) = x(−logx)−1. The hope is that as one approaches the situation of the Duffin-
Schaeffer Conjecture (thatis, f(x) = x) onecanseemoreclearly theoutstanding problems.
In addition any subsequent attack on the conjecture can assume that the series (1) is
diverging “slowly” in certain senses. In this paper we make progress towards this goal by
establishing the following result.
Theorem 1. Let ψ be any non-negative arithmetical function and for any c > 0, define
the function f by
c
0 if x = 0,
f (x) = xexp(−c(log(−logx))(loglog(−logx))) if 0 < x < 1,
c 
1 if x ≥ 1.
Then the divergence of (2) with f = f is sufficient to conclude that λ(W(ψ)) = 1.
 c
It is worth pointing out that this strengthening of Theorem HPV is not a consequence of
simply tweaking the approach taken in [6] – see also the remark at the end of §3 in [6]. By
appealing to the Erd¨os-Vaaler Theorem [10] and to [8, Theorem 2] we can assume without
loss of generality throughout the proof that 1/n ≤ ψ(n) ≤ 1/2 whenever ψ(n) 6= 0. In
view of this it suffices to prove the following theorem.
Theorem 2. The Duffin-Schaeffer Conjecture is true for any non-negative arithmetical
function ψ such that the series
∞
ϕ(n)ψ(n)
(3)
nexp(c(loglogn)(logloglogn))
n=16
X
diverges for some c > 0.
2. The basic framework
Gallagher [4] (see also [5, §2.2]) proved that there is a “zero-one” law for Question 1.
That is, for any given function ψ we have λ(W(ψ)) = 0 or 1. We therefore only need to
prove that under our extra divergence hypothesis λ(W(ψ)) > 0. To do this we need the
following result [5, Lemma 2.3]whose proofinvolves littlemorethanthecorrect application
of the Cauchy-Schwartz inequality.
4 V.BERESNEVICH, G. HARMAN,A. HAYNES,AND S.VELANI
Lemma 1. Let A be a sequence of Lebesgue measurable subsets of R/Z. Let A be the set
n
of α belonging to infinitely many A . Then
n
2 −1
N N
(4) λ(A) ≥ limsup λ(A ) λ(A ∩A ) .
n m m
N→∞ ! !
n=1 m,n=1
X X
The well known Duffin-Schaeffer result [5, Theorem 2.5] toward the Duffin-Schaeffer
Conjecture follows from this lemma together with the elementary bound
(5) λ(E ∩E ) ≤ 8ψ(n)ψ(m) for m 6= n.
m n
However if the sets in the collection {E } were quasi-pairwise independent, i.e. if
n
ϕ(m)ϕ(n)
(6) λ(E ∩E ) ≪ ψ(n)ψ(m) for m 6= n,
m n
m n
then the Duffin-Schaeffer Conjecture would follow at once from (4) together with Gal-
lagher’s zero-one law. Unfortunately (6) does not hold uniformly for all m 6= n. The best
known upper bound for λ(E ∩E ) is the following result.
n m
Lemma 2. For m 6= n we have
(7) λ(E ∩E ) ≪ λ(E )λ(E )P(m,n),
m n m n
where
−1
1
(8) P(m,n) = 1− ,
p
p|mn/Ygcd(m,n)2(cid:18) (cid:19)
p>D(m,n)
with
max(nψ(m),mψ(n))
(9) D(m,n) = .
gcd(m,n)
This was first stated by Strauch [9], but was also given independently by Pollington and
Vaughan [8]. The proof is still essentially elementary, but fairly complicated, and needing
a simple sieve upper bound. Effectively the same result was given earlier by Erd¨os [3].
Clearly what needs to be done in applying Lemma 1 is to show that the factor P(m,n) is
bounded on average.
It is worth pausing here to see what the real difficulties are in estimating λ(E ∩ E ).
m n
Two intervals from E and E overlap if
m n
a b ψ(m) ψ(n)
− < + .
m n m n
(cid:12) (cid:12)
(cid:12) (cid:12)
We lose nothing in terms of the(cid:12)order of(cid:12)magnitude of the bound in replacing this with
(cid:12) (cid:12)
(10) |an−bm| < A(m,n) := 2max(mψ(n),nψ(m)).
DUFFIN-SCHAEFFER WITH EXTRA DIVERGENCE II 5
The length of the intersection is no more than the smallest length of the two intervals
(again nothing is lost in order of magnitude in making this assumption). We thus have
ψ(m) ψ(n)
λ(E ∩E ) ≤ 2min , Σ(m,n),
m n
m n
(cid:18) (cid:19)
where Σ(m,n) denotes the number of solutions to (10) with
1 ≤ a < m, 1 ≤ b < n, gcd(a,m) = 1, gcd(b,n) = 1.
We thus need to show that, at least on average over m,n, we have
ϕ(m)ϕ(n)
Σ(m,n) ≪ A(m,n) .
m n
Now if gcd(m,n) = 1 and A(m,n) is not too small there is no problem with this. The
trouble essentially comes when
A(m,n)
1 ≤ < log(mn).
gcd(m,n)
In that case we are not averaging over enough values of h in the equation an−bm = h to
get the required bound. This is a real problem, not just a deficiency in our knowledge. Our
hope would be that the values of m and n concerned do not make the major contribution
to
λ(E ∩E ).
m n
1≤m,n≤N
X
3. Proof of Theorem 2
Without loss of generality assume that ψ(n) ≥ n−1 whenever ψ(n) 6= 0 – see the discus-
sion just before the statement of the theorem. We divide the integers n into blocks
24h ≤ n < 24h+1.
It then follows that the series (3) diverges with n restricted to blocks with either all the
h even, or all the h odd. Without loss of generality we suppose the series diverges over
blocks with h even, and that ψ(n) = 0 for all integers n which lie in blocks with h odd.
We then note that if m < n, if ψ(m),ψ(n) > 0, and if m and n are in different blocks then
A(m,n) ≥ 2nψ(m) ≥ 2nm−1 ≥ 2ngcd(m,n)m−2 ≫ gcd(m,n)(lognm).
Hence P(m,n) ≪ 1 if m and n belong to different blocks.
Now we consider a block 24h = X ≤ m,n < X4. Write R = loglogX and
ϕ(m)ϕ(n)
Ψ(X) = ψ(n)ψ(m) .
m n
X≤m,n<X4
X
6 V.BERESNEVICH, G. HARMAN,A. HAYNES,AND S.VELANI
By one of Mertens’ theorems, we have for D(m,n) ≥ 1 that
1 R
P(m,n) ≪ exp ≪ .
 p 1+logD(m,n)
D(m,n)<p<logX
X
 
We let D be the collection of pairs (m,n) such that ej ≤ D(m,n) < ej+1. The idea is
j
going to be to divide each ψ(n) by a suitable factor (say ek) so that the contribution from
R consecutive ranges for which we cannot assume P(m,n) ≪ 1 is not of a larger magnitude
than the expected overall contribution. We have
R ϕ(m)ϕ(n)
ψ(n)ψ(m) ≪ RΨ(X)logK,
j +1−k m n
kX≤Kk≤Xj≤k+R (mX,n)∈Dj
since each set D is counted with weight
j
R
≤ ≪ RlogK.
j +1−k
k≤min(j,K)
X
We can therefore choose an integer k ≤ cRlogR such that
R ϕ(m)ϕ(n)
ψ(n)ψ(m) ≪ Ψ(X).
j +1−k m n
k≤Xj≤k+R (mX,n)∈Dj
With this choice of k write
E = D .
j
k≤j≤k+R
[
Now put
ρ(n) = ψ(n)e−k for X ≤ n < X4,
and note that
ek ≤ exp(c(loglogX)(logloglogX)).
We now assume the above procedure has been carried out on all blocks with h even,
and we put ρ(n) = 0 for n in a block with odd h, and we consider the sets E (ρ). By
n
construction
∞
ρ(n)ϕ(n)
= ∞,
n
n=1
X
and
λ(E (ρ)∩E (ρ)) ≪ λ(E (ρ))λ(E (ρ))
m n m n
unless (m,n) ∈ E. But now we also have that
ϕ(n)ϕ(m) ϕ(n)ϕ(m)
ρ(m)ρ(n) P(m,n) ≪ ρ(m)ρ(n)
n m n m
(m,n)∈ E X≤m,n<X4
X X
DUFFIN-SCHAEFFER WITH EXTRA DIVERGENCE II 7
(we note that P(m,n) here does depend on ρ) and so
2
N N
λ(E (ρ)∩E (ρ)) ≪ λ(E (ρ))
m m n
!
m,n=1 n=1
X X
for N taking the values 24h+1. By Lemma 1 and Gallagher’s zero-one law we have that
λ(W(ρ)) = 1, and since W(ρ) ⊆ W(ψ) the proof is completed.
Acknowledgements. AH was supported by EPSRC’s grant EP/F027028/1. SV was sup-
ported by by EPSRC’s grants EP/E061613/1 and EP/F027028/1. SV would like to thank
Andy Pollington for themany discussions centered around theDuffin-Schaeffer Conjecture.
Also a great thanks to Emma Robertson and Kevin Hall for coaching the girls U9’s Wigton
Moor football team to the West Riding girls league and cup double and for putting up with
the dynamo duo Ayesha and Iona. I am truly impressed by your dedication, your fairness
and above all your friendship and openness with the team and parents. Thankyou!
References
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