Relations between exceptional sets for additive problems

0.28 MB·English

by  Koichi Kawada

RELATIONS BETWEEN EXCEPTIONAL SETS FOR
ADDITIVE PROBLEMS
0
1
∗
0 KOICHI KAWADA AND TREVOR D. WOOLEY
2

n Abstract. Wedescribeamethodforboundingthesetofexceptionalinte-
a
gersnotrepresentedbyagivenadditiveformintermsoftheexceptionalset
J
correspondingtoasubform. Illustratingourideaswithexamplesstemming
4
1 from Waring’s problem for cubes, we show, in particular, that the number
of positive integers not exceeding N, that fail to have a representation as

] the sum of six cubes of natural numbers, is O(N3/7).
T
N
.
h
t
a 1. Introduction
m
[ Bounds on exceptional sets in additive problems can oftentimes be improved

1 by replacing a conventional applicationofBessel’s inequality with anargument
v based on the introduction of an exponential sum over the exceptional set, and
5
a subsequent analysis of auxiliary mean values involving the latter generating
9
4 function. Suchastrategyunderliestheearlierworkconcerningslimexceptional
2
sets in Waring’s problem due to one or both of the present authors (see [18],
.
1
[19], [20], [8], [21]). Exponentialsumsoversetsdefiningtheadditiveproblemat
0
0 hand are intrinsic to the application of the Hardy-Littlewood (circle) method
1 thatunderpinssuchapproaches. Onethereforeexpectseachapplicationofsuch
:
v a method to be highly sensitive to the specific identity of the sets in question.
i
X Our goal in this paper is to present an approach which, for many problems,
r is relatively robust to adjustments in the identity of the underlying sets. We
a
illustrateourconclusionswithsomeconsequences forWaring’sproblem, paying
attention in particular to sums of cubes.
In order to present our conclusions in the most general setting, we must
introduce some notation. When N, we write for the complement N
of within N. When a and b aCre⊆non-negative iCntegers, it is convenient\tCo
C
denote by ( )b the set (a,b], and by b the cardinality of (a,b]. Next,
when , C Na, we defiCn∩e |C|a C∩
C D ⊆
= c d : c and d .
C ±D { ± ∈ C ∈ D}
As usual, we use h to denote the h-fold sum + + . Also, we define
D D ··· D
Υ( , ;N) to be the number of solutions of the equation
C D
c d = c d , (1.1)
1 1 2 2
− −
2000 Mathematics Subject Classification. 11P05,11P55,11B75.
Key words and phrases. Exceptional sets, Waring’s problem, Hardy-Littlewood method.
∗
The second author is supported by a Royal Society Wolfson Research Merit Award.
1
2 K. KAWADA ANDT. D. WOOLEY
with c ,c ( )3N and d ,d ( )N. The starting point for our analysis of
1 2 ∈ C 2N 1 2 ∈ D 0
exceptional sets is the inclusion
+ N . (1.2)
A B−B ∩ ⊆ A
In 2 we both justify this (cid:0)trivial relat(cid:1)ion, and also apply it to establish a
§
relation between the cardinalities of complements of sets that encapsulates the
key ideas of this paper. The following theorem is a special case of Theorem
2.1 below, in which we obtain a conclusion with the sets in question restricted
to collections of residue classes.
Theorem 1.1. Suppose that , N. Then for each natural number N, one
A B ⊆
has
2
N + 3N 6 3N Υ( + , ;N).
|B|0 A B 2N A N A B B
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
The conclusion of Th(cid:12)eorem(cid:12)1.1 is no(cid:12)t p(cid:12)articularly transparent, so it seems
3N
appropriate to outline its significance and implications. Note first that
A N
counts the number of natural numbers in the interval (N,3N] that do not lie
(cid:12) (cid:12)
in , which is to say, the exceptional set corresponding to ( )3N. Like(cid:12)wi(cid:12)se,
A A N
3N
we see that + counts the number of natural numbers in the interval
A B 2N
(2N,3N] that do not lie in + , and hence the exceptional set corresponding
(cid:12) (cid:12) A B
to ( + )3N(cid:12). Obse(cid:12)rve next that in many situations of interest, it is possible
A B 2N
to show that the number of solutions c,d of the equation (1.1), counted by
Υ( , ;N), is essentially dominated by the diagonal contribution with c = c
1 2
C D
and d = d . Thus, under suitable circumstances, one finds that
1 2
Υ( + , ;N) + 3N N,
A B B ≪ A B 2N |B|0
and then Theorem 1.1 delivers the bound(cid:12) (cid:12)
(cid:12) (cid:12)
+ 3N 3N / N.
A B 2N ≪ A N |B|0
Inthisway, weareabletos(cid:12)howth(cid:12)atthee(cid:12)xc(cid:12)eptionalsetcorrespondingto + ,
(cid:12) (cid:12) (cid:12) (cid:12) A B
in the interval (2N,3N], is smaller than that corresponding to , in (N,3N],
A
by a factor O(1/ N). With few exceptions, the scale of this improvement is
|B|0
well beyond thecompetence of more classical applications of thecircle method.
ThemostimmediateconsequencesofTheorem1.1concernadditiveproblems
involving squares or cubes. We begin with a cursory examination of the former
problems in 2. It is convenient, when k is a natural number, to describe a
subset of N§as being a high-density subset of the kth powers when (i) one has
nQk : n N , and (ii) for each positive number ε, whenever N is a natural
Q ⊆ { ∈ }
number sufficiently large in terms of ε, then N > N1/k−ε. Also, when θ > 0,
we shall refer to aset N ashaving compl|eQm|0entary density growth exponent
R ⊆
smaller than θ when there exists a positive number δ with the property that,
for all sufficiently large natural numbers N, one has N < Nθ−δ.
R 0
Theorem 1.2. Let be a high-density subset of the s(cid:12)qu(cid:12)ares, and suppose that
N has complemSentary density growth exponent(cid:12) sm(cid:12) aller than 1. Then,
A ⊆
EXCEPTIONAL SETS 3
whenever ε > 0 and N is a natural number sufficiently large in terms of ε, one
has
+ 3N Nε−1/2 3N .
A S 2N ≪ A N
In 2 we provide a sligh(cid:12)tly mo(cid:12)re general con(cid:12)clu(cid:12)sion that captures, inter alia,
§ (cid:12) (cid:12) (cid:12) (cid:12)
the qualitative features of recent work on sums of four squares of primes (see
[18], [6]). Following a consideration of some auxiliary mean values in 3, we
§
advance in 4 to a discussion of additive problems involving cubes.
§
Theorem 1.3. Let be a high-density subset of the cubes, and suppose that
N has complemCentary density growth exponent smaller than θ, for some
A ⊆
positive number θ. Then, whenever ε > 0 and N is a natural number suffi-
ciently large in terms of ε, one has the following estimates:
(a)(exceptional set estimates for + )
A C
3
+ 3N Nε−1/6 3N +Nε−2 3N ;
A C 2N ≪ A N A N
(cid:12)(cid:12) + (cid:12)(cid:12)3N Nε−1/3(cid:12)(cid:12) (cid:12)(cid:12)3N +Nε−1(cid:16)(cid:12)(cid:12) (cid:12)(cid:12)3N(cid:17)2;
A C 2N ≪ A N A N
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(b)(exceptional set estimates for +2 )
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
A C
2
+2 3N Nε−1/2 3N +Nε−4/3 3N , provided that θ 6 1;
A C 2N ≪ A N A N
(cid:12) +2 (cid:12)3N Nε−2/3(cid:12) (cid:12)3N , provided(cid:16)t(cid:12)ha(cid:12)t θ(cid:17)6 13;
(cid:12)A C(cid:12)2N ≪ (cid:12)A(cid:12)N (cid:12) (cid:12) 18
(c)(e(cid:12)xception(cid:12)al set estimat(cid:12)es(cid:12)for +3 )
(cid:12) (cid:12) (cid:12) (cid:12) A C
2
+3 6N Nε−5/3 6N , provided that θ 6 1;
A C 4N ≪ A N
(cid:12) +3 (cid:12)6N Nε−5/6(cid:16)(cid:12) 6(cid:12)N ,(cid:17)provided that θ 6 8.
(cid:12)A C(cid:12)4N ≪ A(cid:12) N(cid:12) 9
The boun(cid:12)ds suppl(cid:12)ied by Theore(cid:12)m(cid:12)1.3(a) have direct consequences for the
(cid:12) (cid:12) (cid:12) (cid:12)
exceptional set in Waring’s problem for sums of cubes. When s is a natural
number and N is positive, write E (N) for the number of positive integers not
s
exceeding N that fail to be represented as the sum of s positive integral cubes.
Thus, if we define = n3 : n N , then we have E (N) = s N. In 5 we
C { ∈ } s C 0 §
establish the following estimates for E (N).
s (cid:12) (cid:12)
(cid:12) (cid:12)
Theorem 1.4. Let τ be any positive number with τ−1 > 2982 + 56√2833.
Then one has
E (N) N37/42−τ, E (N) N5/7−τ, E (N) N3/7−2τ.
4 5 6
≪ ≪ ≪
The estimate presented here for E (N) is simply a restatement of Theorem
4
1.3 of Wooley [17], itself only a modest improvement on Theorem 1 of Bru¨dern
[2]. Our bound for E (N) may be confirmed by a classical approach employing
5
Bessel’s inequality, andindeed such aboundisreportedinequation (1.3)of[3].
Our approach in this paper is simply to apply the first estimate of Theorem
1.3(a). Finally, the estimate for E (N) provided by Theorem 1.4 is new, and
6
may be compared with the bound E (N) N23/42 reported in equation (1.3)
6
≪
4 K. KAWADA ANDT. D. WOOLEY
of [3]. Note that 23 > 0.5476, whereas one may choose a permissible value of
42
τ so that 3 2τ < 0.4283. Of course, in view of Linnik’s celebrated work [10],
7 −
one has E (N) 1 for s > 7.
s
≪
An important strength of Theorem 1.3 is the extent to which it is robust to
adjustments in the set of cubes to which it is applied. It is feasible, for exam-
C
ple, to extract estimates for exceptional sets in the Waring-Goldbach problem
for cubes. The complications associated with inherent congruence conditions
are easily accommodated by simple modifications of our basic framework. In
order to illustrate such ideas, when s is a natural number and N is positive,
write (N) for the number of even positive integers not exceeding N, and
6
E
not congruent to 1 (mod 9), which fail to possess a representation as the
±
sum of 6 cubes of prime numbers. In addition, write (N) for the number
7
E
of odd positive integers not exceeding N, and not divisible by 9, which fail to
possess a representation as the sum of 7 cubes of prime numbers, and denote
by (N) the number of even positive integers not exceeding N that fail to
8
E
possess a representation as the sum of 8 cubes of prime numbers. A discussion
of the necessity of the congruence conditions imposed here is provided in the
preamble to Theorem 1.1 of [19]. By applying a variant of Theorem 1.3, in 5
§
we obtain the following upper bounds on (N) (6 6 s 6 8).
s
E
Theorem 1.5. One has
(N) N23/28, (N) N23/42 and (N) N3/14.
6 7 8
E ≪ E ≪ E ≪
For comparison, Theorem 1 of Kumchev [9] supplies the weaker bounds
(N) N31/35, (N) N17/28 and (N) N23/84.
6 7 8
E ≪ E ≪ E ≪
We have more to say concerning the Waring-Goldbach problem, so we defer
further consideration of allied conclusions to a future occasion.
As the final illustration of our methods, in 6 we consider Waring’s problem
§
for biquadrates. Since fourth powers are congruent to 0 or 1 modulo 16, a
sum of s biquadrates must be congruent to r modulo 16, for some integer
r satisfying 0 6 r 6 s. If n is the sum of s < 16 biquadrates and 16 n,
|
moreover, then n/16 is also the sum of s biquadrates. It therefore makes
sense, in such circumstances, to consider the representation of integers n with
n r (mod 16) for some integer r with 1 6 r 6 s. Define Y (N) to be the
s
≡
number of integers n not exceeding N that satisfy the latter condition, yet
cannot be written as the sum of s biquadrates.
Theorem 1.6. Write δ = 0.00914. Then one has
Y (N) N15/16−δ, Y (N) N7/8−δ, Y (N) N13/16−δ,
7 8 9
≪ ≪ ≪
Y (N) N3/4−2δ, Y (N) N5/8−2δ.
10 11
≪ ≪
Here, the estimates for Y (N) when 7 6 s 6 9 follow from a classical ap-
s
plication of Bessel’s inequality, combined with the work of Vaughan [12] and
Bru¨dern and Wooley [4] concerning sums of biquadrates. Such techniques
would also yield the bounds Y (N) N3/4−δ and Y (N) N11/16−δ, each
10 11
≪ ≪
EXCEPTIONAL SETS 5
of which is inferior to the relevant conclusion of Theorem 1.6. We remark that
superior estimates are available if one is prepared to omit the congruence class
s modulo 16, or s 1 and s modulo 16, from the integers under consideration
−
for representation as the sum of s biquadrates. We refer the reader to Theo-
rems 1.1 and 1.2 of the authors’ earlier work [8] for details. Finally, we note
that in view of Theorem 1.2 of Vaughan [12], one has Y (N) 1 for s > 12.
s
≪
In 7 we discuss further the abstract formulation of exceptional sets un-
§
derlying Theorem 1.1, and consider the consequences of the most ambitious
conjectures likely to hold for the additive theory of exceptional sets.
Throughout, theletter εwill denote asufficiently small positive number. We
use and to denote Vinogradov’s well-known notation, implicit constants
≪ ≫
depending at most on ε, unless otherwise indicated. In an effort to simplify
our analysis, we adopt the convention that whenever ε appears in a statement,
then we are implicitly asserting that for each ε > 0, the statement holds for
sufficiently large values of the main parameter. Note that the “value” of ε
may consequently change from statement to statement, and hence also the
dependence of implicit constants on ε.
2. The basic inequality
Our goal in this section is to establish the upper bound presented in The-
orem 1.1, illustrating this relation with the inexpensive conclusion recorded
in Theorem 1.2. We begin by spelling out the inclusion (1.2). The proof
is by contradiction. Let n + and b . Suppose, if possible, that
∈ A B ∈ B
n b . Then there exists an element a of for which n b = a, whence
− ∈ A A −
n = a+b + . But then n + , contradicting our initial hypothesis.
We are the∈reAforeBforced to conclu6∈dAe thaBt n b , so that if n b N, then
n b . In this way, we confirm that −+ 6∈ A N , −as d∈esired.
− ∈ A A B −B ∩ ⊆ A
We establish Theorem 1.1 in a more general form useful in applications. In
(cid:0) (cid:1)
this context, when q is a natural number and a 0,1,...,q 1 , we define
∈ { − }
= by
a a,q
P P
= a+mq : m Z .
a,q
P { ∈ }
Also, we describe a set as being a union of arithmetic progressions modulo
q when, for some subsetLL of 0,1,...,q 1 , one has
{ − }
= .
l,q
L P
l∈L
[
In such circumstances, given a subset of N and integers a and b, it is conve-
C
nient to write
b = min b .
hC ∧Lia l∈L |C ∩Pl,q|a
Theorem 2.1. Suppose that , N. In addition, let , and be
A B ⊆ L M N
unions of arithmetic progressions modulo q, for some natural number q, and
suppose that + . Then for each natural number N, one has
N ⊆ L M
2
N + 3N 6 q 3N Υ( + , ;N).
hB∧Li0 A B∩N 2N A∩M N A B ∩N B∩L
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
6 K. KAWADA ANDT. D. WOOLEY
Proof. We begin by deriving a variant of the relation (1.2). Let N be a
large natural number, and suppose that , , satisfy the hypotheses of
the statement of the theorem. We may supLpoMse tNhat there are sets A,B,C
⊆
0,1,...,q 1 with the property that
{ − }
= , = and = .
a b c
M P L P N P
a∈A b∈B c∈C
[ [ [
Moreover, in view of the hypothesis + , there exists a subset D of
B A, with card(D) 6 q, satisfying Nthe⊆prLoperMty that
×
= ( + ).
c a b
P P P
c∈C (b,a)∈D
[ [
In particular, for each c C, there exists a pair (b,a) D satisfying the
∈ ∈
property that = + .
c a b
P P P
Suppose now that c C, and that (b,a) D satisfies the condition that
∈ ∈
= + . Let N be a large natural number, and suppose that n
c a b
P P P ∈
+ 3N and b ( )N. Then
A B∩Pc 2N ∈ B∩Pb 0
(cid:0) (cid:1)
n b (N,3N] ( ) = (N,3N] ,
c b a
− ∈ ∩ P −P ∩P
and so the argument in the opening paragraph of this section shows that
3N
n b . We therefore deduce that
a
− ∈ A∩P N
(cid:0) (cid:1)
+ 3N ( )N 3N , (2.1)
A B ∩Pc 2N − B ∩Pb 0 ⊆ A∩Pa N
(cid:0) (cid:1) (cid:0) (cid:1)
whence
3N > card + 3N ( )N .
A∩Pa N A B ∩Pc 2N − B∩Pb 0
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:0) (cid:1)
Next, write(cid:12) ρ (m)(cid:12)for the number of solutions of the equation m = n b,
bc
−
3N
with n + and b ( )N. An application of Cauchy’s
∈ A B ∩Pc 2N ∈ B ∩ Pb 0
inequality shows that
(cid:0) (cid:1)
2
ρ (m) 6 1 ρ (m)2 . (2.2)
bc bc
(cid:16)Xc∈C 16Xm63N (cid:17) (cid:16)Xc∈C 16Xm63N (cid:17)(cid:16)Xc∈C 16Xm63N (cid:17)
ρbc(m)>1
On recalling the definition of Υ( , ;N) from the preamble to Theorem 1.1,
C D
we have
ρ (m)2 = Υ( + , ;N)
bc c b
A B∩P B∩P
c∈C 16m63N c∈C
X X X
6 Υ( + , ;N).
A B∩N B∩L
EXCEPTIONAL SETS 7
Moreover, a moment’s reflection confirms that
ρ (m) = 1
bc
Xc∈C 16Xm63N Xc∈C n∈(A+XB∩Pc)32NN b∈(BX∩Pb)N0
> min N 1
b
b∈B |B ∩P |0
n∈(A+XB∩N)32NN
= N + 3N .
hB∧Li0 A B ∩N 2N
Observe next that, in view of the relation ((cid:12)2.1), when ρ(cid:12)bc(m) > 1, one has
3N (cid:12) (cid:12)
m . Thus one has the upper bound
a
∈ A∩P N
3N
(cid:0) (cid:1) 1 6 1 6 ,
A∩M N
1ρ6bcXm(m6)3>N1 m∈(AX∩Pa)3NN (cid:12)(cid:12) (cid:12)(cid:12)
whence
3N
1 6 q .
A∩M N
c∈C 16m63N
XρbcX(m)>1 (cid:12)(cid:12) (cid:12)(cid:12)
Theconclusion ofthetheoremfollowsonsubstituting theserelationsinto (2.2).
(cid:3)
The conclusion of Theorem 1.1 is immediate from the case q = 1 of Theorem
2.1, in which , and are each taken to be Z. As a first illustration of the
L M N
ease with which Theorems 1.1 and 2.1 may be applied to concrete problems,
we now establish a theorem which implies Theorem 1.2 by using the strategy
presented first in the proof of Theorem 1.1 of [18]. We first extend the notation
introduced in the preamble to the statement of Theorem 1.2. Let be a union
L
of arithmetic progressions modulo q, for some natural number q. When k is
a natural number, we describe a subset of N as being a high-density subset
of the kth powers relative to when (i)Qone has nk : n N , and (ii)
L Q ⊆ { ∈ }
for each positive number ε, whenever N is a natural number sufficiently large
in terms of ε, then N N1/k−ε. Also, when θ > 0, we shall refer
to a set N as hhaQvin∧gLi0-co≫mqplementary density growth exponent smaller
R ⊆ L
than θ when there exists a positive number δ with the property that, for all
N
sufficiently large natural numbers N, one has < Nθ−δ.
R∩L 0
Theorem 2.2. Let , and be unions of(cid:12)arithm(cid:12)etic progressions modulo
L M N (cid:12) (cid:12)
q, for some natural number q, and suppose that + . Suppose also
that is a high-density subset of the squares relatiNve t⊆o L, anMd that N has
S L A ⊆
-complementary density growth exponent smaller than 1. Then, whenever
M
ε > 0 and N is a natural number sufficiently large in terms of ε, one has
+ 3N Nε−1/2 3N .
A S ∩N 2N ≪q A∩M N
Proof. Throughout(cid:12)the proof of(cid:12)this theorem,(cid:12)implicit(cid:12)constants may depend
(cid:12) (cid:12) (cid:12) (cid:12)
on q. Let N be a large natural number, and suppose that , , satisfy
L M N
the hypotheses of the statement of the theorem. Also, let be a high density
S
subset of the squares relative to . Then, in particular, there is a subset
L T
8 K. KAWADA ANDT. D. WOOLEY
of N for which = n2 : n . Consider also a subset of N having
S ∩L { ∈ T} A
-complementary density growth exponent smaller than 1. Write P = [N1/2].
M
The quantity Υ( + , ;N) counts the number of solutions of the
A S ∩N S ∩L
equation
n n = x2 x2, (2.3)
1 − 2 1 − 2
with n ,n + 3N and x ,x ( )P. There are plainly
1 2 ∈ A S ∩N 2N 1 2 ∈ T 0
(cid:0) (cid:1) + 3N P
A S ∩N 2N |T|0
solutions of this equation wit(cid:12)h n = n a(cid:12)nd x2 = x2. Given any one of the
(cid:12) 1 2 (cid:12) 1 2
2
3N
O +
A S ∩N 2N
(cid:16)(cid:16) (cid:17) (cid:17)
(cid:12) (cid:12)
available choices of n1 and n2 w(cid:12) ith n1 = n(cid:12)2, meanwhile, one may apply an
6
elementary estimate for the divisor function to show that there are O(Nε)
possible choices for x x and x +x satisfying (2.3), whence also for x and
1 2 1 2 1
−
x . On noting that P = N, we find that
2 |T|0 |S ∩L|0
Υ( + , ;N) + 3N N
A S ∩N S ∩L ≪ A S ∩N 2N |S ∩L|0
2
(cid:12)+Nε +(cid:12) 3N .
(cid:12) (cid:12)
A S ∩N 2N
(cid:16) (cid:17)
(cid:12) (cid:12)
We substitute this last estimate into the co(cid:12)nclusion of (cid:12)Theorem 2.1, and
thereby deduce that
N 2 + 3N 3N N
hS ∧Li0 A S ∩N 2N ≪ A∩M N |S ∩L|0
(cid:0) (cid:1) (cid:12) (cid:12) (cid:12)+Nε (cid:12) 3N + 3N .
(cid:12) (cid:12) (cid:12) A(cid:12)∩M N A S ∩N 2N
But since is a high-density subset of the(cid:12)squares(cid:12)rela(cid:12)tive to , a(cid:12)nd the set
S (cid:12) (cid:12) (cid:12) L (cid:12)
has -complementary density growth exponent smaller than 1, then there
A M
exists a positive number δ with the property that
Nδ 3N N1−δ < (N1/2−ε)2 N 2.
A∩M N ≪ ≪ hS ∧Li0
In addition, one(cid:12)has (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
N
|S ∩L|0 Nε−1/2.
( N)2 ≪
hS ∧Li0
Thus we deduce that
+ 3N Nε−1/2 3N +Nε−δ + 3N ,
A S ∩N 2N ≪ A∩M N A S ∩N 2N
and the(cid:12)conclusion o(cid:12)f the theorem(cid:12)follows(cid:12)at once. (cid:12) (cid:12) (cid:3)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
The estimate claimed in Theorem 1.2 follows from Theorem 2.2 on putting
q = 1 and taking , and each to be Z.
L M N
EXCEPTIONAL SETS 9
3. Auxiliary mean values involving cubes
Before applying Theorems 1.1 or 2.1 to additive problems involving cubes, it
is necessary toestablish someauxiliary mean valueestimates in order tobound
the expression Υ( + , ;N) relevant to our problems. This we accomplish
A B B
in the present section.
Let and beunionsofarithmeticprogressionsmoduloq, forsomenatural
L N
number q. In addition, let bea high-density subset of thecubes relative to ,
and let be a subset of N.CConsider a large natural number N, and write PL=
N1/3. OAbserve first that when s N, the quantity Υ( +s ,s( );N)
∈ A C∩N C∩L
is bounded above by the number of solutions of the equation
s
n n = (x3 y3), (3.1)
1 − 2 i − i
i=1
X
3N
with n ,n +s and 1 6 x ,y 6 P (1 6 i 6 s). Write (N) for
1 2 ∈ A C ∩N 2N i i Z
3N
+s and Z for card( (N)). Also, define the exponential sums
(cid:0) (cid:1)
A C ∩N 2N Z
(cid:0) (cid:1)f(α) = e(αx3) and K(α) = e(nα).
16x6P n∈Z(N)
X X
Here, as usual, we write e(z) for e2πiz. Then, on considering the underlying
diophantine equation, it follows from (3.1) that
1
Υ( +s ,s( );N) 6 f(α)2sK(α)2 dα. (3.2)
A C ∩N C ∩L | |
Z0
We begin by considering the situation in which s = 1.
Lemma 3.1. One has
1
f(α)2K(α)2 dα Pε(P3/2Z +Z5/3)
| | ≪
Z0
and
1
f(α)2K(α)2 dα PZ +P1/2+εZ3/2.
| | ≪
Z0
Proof. We estimate the integral in question first by means of the Hardy-
Littlewood method. When a Z and r N, define the major arcs N(r,a) by
∈ ∈
putting
N(r,a) = α [0,1) : rα a 6 P−2 ,
{ ∈ | − | }
and then take N to be the union of the arcs N(r,a) with 0 6 a 6 r 6 P and
(a,r) = 1. Also, write n = [0,1) N. Next, define Υ(α) for α [0,1) by taking
\ ∈
Υ(α) = (r +P3 rα a )−1,
| − |
when α N(r,a) N, and otherwise by putting Υ(α) = 0. Also, define the
∈ ⊆
function f∗(α) for α [0,1) by taking f∗(α) = PΥ(α)1/3. On referring to
∈
Theorems 4.1 and 4.2, together with Lemma 2.8, of Vaughan [13], one readily
confirms that the estimate
f(α) f∗(α)+P1/2+ε
≪
10 K. KAWADA ANDT. D. WOOLEY
holds uniformly for α N. An application of Weyl’s inequality (see Lemma
∈
2.4 of [13]), meanwhile, reveals that
sup f(α) P3/4+ε.
α∈n | | ≪
Thus we find that, uniformly for α [0,1), one has
∈
f(α) 2 f∗(α)2 +P3/2+ε, (3.3)
| | ≪
whence
1
f(α)2K(α)2 dα P3/2+εI +P2I , (3.4)
1 2
| | ≪
Z0
where
1 1
I = K(α) 2dα and I = Υ(α)2/3 K(α) 2dα.
1 2
| | | |
Z0 Z0
By Parseval’s identity, one has I = Z. Meanwhile, an application of
1
H¨older’s inequality combined with Lemma 2 of [1] shows that
1 2/3 1 1/3
I Υ(α) K(α) 2dα K(α) 2dα
2
≪ | | | |
(cid:16)Z0 (cid:17) (cid:16)Z0 (cid:17)
Pε−3(PZ +Z2) 2/3Z1/3.
≪
On substituting the(cid:0)se estimates into(cid:1)(3.4), we deduce that
1
f(α)2K(α)2 dα P3/2+εZ +Pε(P2/3Z +Z5/3),
| | ≪
Z0
and the first conclusion of the lemma follows.
In order to confirm the second estimate, we begin by considering the un-
derlying diophantine equation. One finds that the mean value in question is
bounded above by Q(m)2, where Q(m) denotes the number of solutions
m
of the equation x3 + n = m, with 1 6 x 6 P and n (N). The desired
P ∈ Z
conclusion therefore follows from a trivial modification of Theorem 1 of [5]
(see, for example, Theorem 6.2 of [13] with k = 3, j = 1 and ν = 1, or the
case k = 3 and j = 1 of Lemma 6.1 below). (cid:3)
Next we consider the situation with s = 2.
Lemma 3.2. One has
1
f(α)4K(α)2 dα P2Z +P11/6+εZ2,
| | ≪
Z0
and
1
f(α)4K(α)2 dα Pε(P5/2Z +P2Z3/2 +PZ2).
| | ≪
Z0
Proof. On making use of a bound of Parsell based on the methods of Hooley
(see Lemma 2.1 of [11], and also [7]), the argument of the proof of Lemma 10.3
of [19] supplies the first bound claimed in the lemma. We refer the reader to
the discussion on pages 420 and 447 of [19] for amplification on this matter.