1. Introduction. In 1989, Fouvry and Iwaniec [4] used the double large sieve inequality due to Bombieri and Iwaniec [1] to estimate multiple expo- nential sums with monomials

XCII.3 (2000)

Multiple exponential sums with monomials

by

Xiaodong Cao (Beijing) and Wenguang Zhai (Jinan)

1. Introduction. In 1989, Fouvry and Iwaniec [4] used the double large sieve inequality due to Bombieri and Iwaniec [1] to estimate multiple expo- nential sums with monomials

X

m1∼M1

. . . X

mk∼Mk

ϕ

. . . ϕ

e



T m

. . . m

M

. . . M

 ,

where e(θ) := e

, α

∈ R, ϕ

∈ C. This method is often superior to the classical methods (for comparison, see [5]–[7], [10] for example).

Let A be the A-process of Weyl–van der Corput and D the process of applying the large sieve inequality. Then the method of Fouvry and Iwaniec consists in an application of the AD process, which ends with a spacing problem for the points t(m, q) and gives rise to a lot of applications. This method was sharpened in Liu [8]. Recently, this spacing problem was further improved in Sargos and Wu [9] by an ingenious new idea.

But sometimes if T is very large, one should use the A

D process (i.e.

two times the A-process followed by the D-process), which naturally involves the spacing problem for the points t(m, q

, q

) := t(m+q

, q

)−t(m−q

, q

) for m ∼ M , q

∼ Q

and q

∼ Q

. Recently, the authors [2] studied the spacing problem of t(m, q

, q

) with q

fixed by the method of Fouvry and Iwaniec [4], and used the result to study some problems in number theory (see Cao and Zhai [2], [3]). This idea plays a key role in the two papers.

Our aim is to give better results on the spacing of t(m, q

, q

). In Sec- tion 2, some preliminary lemmas are given. In Section 3, we use the method of Fouvry and Iwaniec to study the spacing of t(m, q

, q

) for all m, q

, q

, which can be used to deal with the case of q

near to q

in applications.

In Section 4 we use the new idea of Sargos and Wu [9] and the method in

2000 Mathematics Subject Classification: Primary 11L07; Secondary 11N25, 11N45.

This work is supported by the National Natural Science Foundation of China (Grant No. 19801021) and Natural Science Foundation of Shandong Province (Grant No.

Q98A02210).

[195]

Section 3 to study the same spacing problem, which is used in application for q

larger than q

. The spacing problems for other related points are con- sidered in Section 5. In Section 6, some estimates for exponential sums with monomials are given. Applications of these results will be given elsewhere.

Notations. m ∼ M means that M < m ≤ 2M ; f  g means f  g  f ; ktk := min

|n−t| and ψ(t) := t−[t]−1/2 for real t. We also use ε to denote an arbitrarily small positive constant, ε

to denote an unspecified constant multiple of ε, and ε

to denote a fixed suitably small positive number. We also use the notations

C

:= α(α − 1) . . . (α − m + 1)

m! and C

= 1.

Throughout the paper, we always set (for α 6= 0, 1, 2, 3) t(m, q) := t(m, q; α) = (m + q)

− (m − q)

,

t(m, q

, q

) := t(m, q

, q

; α) = t(m + q

, q

) − t(m − q

, q

).

2. Some preliminary lemmas. Let δ > 0, M ≥ 1 and f ∈ C[M, 2M ].

Define

(2.1) R(f, δ) := |{m ∈ [M, 2M ] : kf (m)k < δ}|,

which denotes the number of integer points in the δ-neighbourhood of f (t) for M ≤ t ≤ 2M . In this section all constants implied by “”, “O” and

“” may depend only on α and β.

We set h := (n, e n, q, e q) and H := (N, Q). We write h ∼ H for n, e n ∼ N , q, e q ∼ Q and define H := {h : h ∼ H}. The following lemma is Theorem 2 of Sargos and Wu [9].

Lemma 2.1. Let α, β ∈ R with αβ 6= 0 and let M ≥ 1, N ≥ 1, Q ≥ 1 and L := log(2M N Q). Let H

⊆ H and g

(t) ∈ C

[M, 2M ] with g

(t)  µM , g

(t)  µ for t ∼ M , h ∈ H

. Put f

(t) := v(h)t + g

(t) with v(h) :=

e

n

q e

/(n

q

). Then X

h∈H^∗

R(f

, δ)  δM |H

| + |H

| + (δN Q|H

|µ

L)

(2.2)

+ µ

M |H

| + (δ

N

Q

|H

|

µ

)

+ N Q(δM |H

|)

.

Lemma 2.2. Let α(α − 1)(α − 2)(α − 3) 6= 0, α ∈ R and q

, q

> 0. Then for |u| ≤ 1/(10(q

+ q

)), J ∈ N, we have

σ(u) = σ(u, q

, q

) :=

 t(1, q

u, q

u; α) 4α(α − 1)q

q

u



(2.3)

= 1 + X

σ

(q

, q

)u

+ O

(((q

+ q

)u)

),

where

σ

(q

, q

) = X

(SC)

k

!

k

!k

! . . . k

! · C

(2C

)

(2.4)

×



Y

i=2

 C

X

C

q

q





: = X

a

q

q

(a

= a

, a

is real),

and where SC denotes the following conditions: k

≥ 0 (i = 1, . . . , J + 1), k

+ 2k

+ . . . + Jk

= j, k

+ k

+ . . . + k

= k

.

In particular

σ

= 2C

C

C

(q

+ q

) := C(q

+ q

), (2.5)

σ

= 4C

(C

)

(C

)

(q

+ q

)

(2.6)

+ C

C

C

(3q

+ 10q

q

+ 3q

).

P r o o f. For µ 6= 0 and |x| < 1/2 we have

(2.7) (1 + x)

=

X

C

x

.

From (2.7) one easily gets t(1, q

u, q

u; α) =

X

C

{((q

+ q

)u)

− (−(q

− q

)u)

} (2.8)

+ X

C

{−((q

− q

)u)

+ (−(q

+ q

)u)

}

= 2 X

C

{(q

+ q

)

− (q

− q

)

}u

. It is well known that

(2.9) (x

+ . . . + x

)

= X

k₁+...+k_m=n k_i≥0, i=1,...,m

n!

k

! . . . k

! x

. . . x

.

It follows from (2.9) that

(2.10) (q

+ q

)

− (q

− q

)

= X

C

q

q

− X

C

q

(−q

)

= 2 X

C

q

q

. Combining (2.8) and (2.10) we obtain

(2.11) σ(u) = (1 + A(q

, q

, u) + O((q

u + q

u)

))

, where

A(q

, q

, u) =

J+1

X

m=2

C

2C

 X

k=1

C

q

q

 u

. Now, using (2.11), (2.7) and the mean-value theorem, we have (2.12) σ(u) = 1 +

J+1

X

k1=1

C

A

(q

, q

, u) + O(((q

+ q

)u)

).

Finally, by (2.12), (2.9) and after a simple calculation, we easily deduce σ(u) = 1 +

J+1

X

k1=1

C

(2C

)

 X

k2+k3+...+kJ+1=k1

k_i≥0, i=2,3,...,J+1

k

! k

!k

! . . . k

!

× n

Y

i=2

 C

X

C

q

q



on

Y

i=2

u

o

+ O(((q

+ q

)u)

),

and from the above expression Lemma 2.2 can be proved at once.

Lemma 2.3. Let α, β ∈ R, α(α − 1)β(αβ − β − 1) 6= 0, r, q > 0 and let γ := 1/(αβ − β − 1). Then for |u| ≤ 1/(10(r + q)), J ∈ N,

σ

(u) = σ

(u, q, r) :=

 t(1, qu, ru; α) 2(α − 1)(2α)

q

ru



(2.13)

= 1 + X

σ

(q, r)u

+ O(((q + r)u)

), where

σ

(q, r) = γ

 4C

α(α − 1) + 6C

αβ + (α − 1)C

β

 r

(2.14)

+ γ

 4C

α(α − 1) + 2C

αβ



q

,

σ

(q, r) = X

b

q

r

, (2.15)

where b

= b

(α, β) is real for all i, j ≥ 0.

P r o o f. The proof is similar to that of Lemma 2.2, so we omit the details.

We notice that the degree of u in the Taylor expansion of σ

(u, q, r) is al- ways even since σ

(−u, q, r) = σ

(u, q, r). Moreover, we have σ

(u, q, −r) = σ

(u, q, r).

3. Spacing problem for the points t(m, q

, q

) (I). This section is devoted to investigating the spacing problem for the points t(m, q

, q

) with q

“near” to q

by the method of Fouvry and Iwaniec [4]. Throughout this and the next section all constants implied by “”, “O” and “” depend at most on α and ε (or ε

); we also use the following notations. Let M ≥ 10, Q

≥ 1, Q

≥ 1, η > 0, δ > 0 and ∆ > 0. We set T := M

Q

Q

and L := log(2M Q

Q

). Therefore for m ∼ M , q

∼ Q

, q

∼ Q

and Q

+ Q

< M/3, we have t(m, q

, q

)  T .

Let F(M, Q

, Q

, ∆) denote the number of sextuplets (m, e m, q

, e q

, q

, e q

) with m, e m ∼ M, q

, e q

∼ Q

and q

, e q

∼ Q

, satisfying

(3.1) |t(m, q

, q

) − t( e m, e q

, e q

)| ≤ ∆T.

Theorem 1. Let 1 ≤ Q

≤ Q

≤ M

. Then

F(M, Q

, Q

, ∆)M

 M Q

Q

+ ∆(M Q

Q

)

(3.2)

+ M

Q

Q

+ Q

Q

. We set

A :=

 q

q

e q

e q



, B := Aσ

(q

, q

) − A

σ

( e q

, e q

),

where σ

(q

, q

) is defined by (2.5). The following lemma is an analogue of Lemma 4 of Fouvry and Iwaniec [4].

Lemma 3.1. Let D(M, Q

, Q

, ∆) denote the number of couples (m, q), q = (q

, e q

, q

, e q

), with m ∼ M ; q

, e q

∼ Q

; q

, e q

∼ Q

, satisfying kAm − Bm

k ≤ ∆. If 1 ≤ Q

≤ Q

≤ M

, then

(3.3) M

D(M, Q

, Q

, ∆)  M Q

Q

+ ∆M (Q

Q

)

+ Q

Q

. P r o o f. Since D(M, Q

, Q

, ∆) is non-decreasing in ∆, we can assume that (Q

Q

)

< ∆ < 1. First we estimate the number of couples (m, q) with |B| ≤ ∆M by a crude argument. In this case we have kAmk ≤ 2∆, which implies

(3.4)

 q

q

e q

q e



− m e m  

   ∆M

.

By Lemma 1 of Fouvry and Iwaniec [4], the number of quadruples (m, e m, n, e n) with m, e m ∼ M , n, e n ∼ Q

Q

such that

 n e n



− m e m  

   ∆M

is

(3.5)  M Q

Q

L + (∆M

)M

(Q

Q

)

= M Q

Q

L + ∆M (Q

Q

)

. Since Q

Q

≤ q

q

, e q

q e

≤ 4Q

Q

, the number of such (m, q) is

(3.6)  M (Q

Q

)

+ ∆M (Q

Q

)

by a divisor argument.

By Lemma 3 of Fouvry and Iwaniec [4] (see also the proof of Lemma 2 in Liu [8]) we find that for S = ∆

, the number of (m, q) with |B| ≥ ∆M is (3.7)  ∆M (Q

Q

)

+ E

+ E

,

where

E

:= E

(M, Q

, Q

, ∆) = ∆ X

1≤s≤S

X

q1,eq1∼Q1

q2,eq2∼Q2

min(M, 1/kAsk), (3.8)

E

:= E

(M, Q

, Q

, ∆) (3.9)

= ∆ X

1≤s≤S

X

q₁,eq₁∼Q₁, q₂,eq₂∼Q₂ kAsk3s|B|M⁻²

min(M, (|B|sM

)

).

We estimate E

first. By a simple splitting argument for the interval [1, ∆

], we get for some 1 ≤ S

≤ ∆

= S,

E

 ∆L X

s∼S₁

X

q1,eq1∼Q1, q2,eq2∼Q2

kAsk≤M⁻¹

min(M, 1/kAsk) (3.10)

+ ∆L X

s∼S1

X

q1,eq1∼Q1, q2,eq2∼Q2

kAsk>M⁻¹

min(M, 1/kAsk)

= ∆LT

+ ∆LT

.

Applying the same argument as for (3.6), we obtain

(3.11) T

 M {S

(Q

Q

)

+ (M

S

)S

(Q

Q

)

}.

To treat T

, applying the splitting argument to the interval [M

, 1], we get for some M

≤ δ ≤ 1,

T

 L δ

X

s∼S1

X

q₁,eq₁∼Q₁, q₂,eq₂∼Q₂ δ<kAsk≤2δ

1  L δ

X

s∼S1

X

q₁,eq₁∼Q₁, q₂,eq₂∼Q₂ δ<kAsk≤2δ

1.

Similarly to the estimate for T

in (3.11), we also have

(3.12) T

 δ

L{S

(Q

Q

)

+ (δS

)S

(Q

Q

)

}.

From (3.10)–(3.12) we obtain

(3.13) E

 M (Q

Q

)

L + (Q

Q

)

L.

To treat E

, we split the range of summation into subsets defined by S

< s ≤ 2S

and R < |B| ≤ 2R, where 1 ≤ S

≤ ∆

and ∆M ≤ R ≤ Q

. Thus for some S

and R,

E

 L

∆ X

s∼S1

X

q₁,eq₁∼Q₁, q₂,eq₂∼Q₂ kAsk3s|B|M⁻², R<|B|≤2R

min(M, (|B|sM

)

) (3.14)

 L

∆ X

s∼S₁

X

q1,eq1∼Q1, q2,eq2∼Q2

kAskS1RM⁻²

min(M, (RS

M

)

)

:= L

∆T

.

If R ≤ M S

, similarly to (3.11) we have

(3.15) ∆T

 ∆{S

M (Q

Q

)

+ S

(Q

Q

)

}.

If R > M S

, similarly to (3.12) we also have

∆T

 L∆(RS

M

)

X

s∼S1

X

q₁,eq₁∼Q₁, q₂,eq₂∼Q₂ kAskS₁RM⁻²

1 (3.16)

 L∆(RS

M

)

{S

(Q

Q

)

+ (RM

)S

(Q

Q

)

}.

Combining (3.14)–(3.16) we get

(3.17) E

 M (Q

Q

)

L + (Q

Q

)

L + (∆M )

(Q

Q

)

Q

L.

Finally, since D(M, Q

, Q

, ∆) is non-decreasing in ∆, we can replace ∆ on the right-hand side of (3.17) by ∆ + M

Q

; this new ∆ and the con- dition Q

≤ Q

≤ M

assure that |B| < ∆M

holds. This completes the proof of Lemma 3.1.

Proof of Theorem 1. Inequality (3.1) implies that

|t

(m, q

, q

) − t

( e m, e q

, e q

)| ≤ ∆T

. By (2.3) with u = 1/m and J = 1, we deduce that

(3.18) |Am + Bm

− e m|  ∆M + Q

M

.

Clearly (3.18) implies kAm + Bm

k  ∆M + Q

M

, and for given

m, q

, e q

, q

, e q

the number of e m is bounded by O(1 + ∆M ). Applying

Lemma 3.1 with ∆ = ∆M + Q

M

we get F(M, Q

, Q

, ∆)  (1 + ∆M )

× (M (Q

Q

)

+ ∆M

(Q

Q

)

+ M

Q

Q

) + (1 + ∆M )Q

Q

L.

If ∆M < 1 we obtain the bound (3.2), otherwise the trivial estimate yields F(M, Q

, Q

, ∆)  (1 + ∆M )M (Q

Q

)

 ∆(M Q

Q

)

.

Now the proof of Theorem 1 is finished.

4. Spacing problem for the points t(m, q

, q

) (II). In this section, we shall use the new idea developed by Sargos and Wu [9] and the method in the proof of Theorem 1 to investigate the spacing problem for the points t(m, q

, q

) for q

“larger” than q

. Before stating our result, we need some notations. For q

∼ Q

, q

∼ Q

, we have (q

q

)

∈ I, where I :=

[c(Q

Q

)

, c

(Q

Q

)

], and c, c

are two suitable positive constants depending on α only. Let I

⊆ I with |I

| = η(Q

Q

)

. We use E(M, Q

, Q

, ∆, I

) to denote the number of sextuplets (m, e m, q

, e q

, q

, e q

) with m, e m ∼ M , q

, e q

∼ Q

, q

, e q

∼ Q

, such that

|t(m, q

, q

) − t( e m, e q

, e q

)| ≤ ∆T, (4.1)

(q

q

)

∈ I

, (e q

q e

)

∈ I

.

Theorem 2. If Q

≥ 1, Q

M

≤ Q

≤ M

and Q

Q

≤ M

, then there exists an

η = η(M, Q

, Q

) ∈

 max

 Q

Q

, 3L

Q

Q

 , c

c − 1



such that

(4.2) η

M

X

0≤k≤K

E(M, Q

, Q

, ∆, I

)

 M Q

Q

+ ∆M

Q

Q

+ (M Q

Q

)

+ M

Q

Q

+ (∆M

Q

Q

)

+ (∆M

Q

Q

)

+ (Q

Q

)

+ (∆M

Q

Q

)

+ (M

Q

Q

)

,

where I

:= [a

, (1 + η)a

], a

:= (1 + η)

c(Q

Q

)

, and K :=

[log(c

/c)/η].

Define

q := (q

, e q

, q

, e q

), v(q) :=

 q

q

e q

e q



,

Q := {q : q

, e q

∼ Q

; q

, e q

∼ Q

},

v

(q) := v(q)σ

(q

, q

) − v

(q)σ

(e q

, e q

),

where σ

(q

, q

) is given by (2.5), and we define ψ

:= ψ

(m, q) by the recurrence relation

ψ

:= v

(q)m

, (4.3)

ψ

= X

{v(q)σ

(q

, q

)m

− σ

(e q

, e q

)(v(q)m + ψ

)

} (4.4)

for J ≥ 2.

Let E

(M, Q

, Q

, δ, I

) be the number of couples (m, q) with m ∼ M, q ∼ Q such that

kv(q)m + ψ

(m, q)k ≤ δ, (4.5)

(q

q

)

∈ I

, (e q

q e

)

∈ I

. Lemma 4.1. If 1 ≤ Q

≤ Q

≤ ε

M , then for any J ≥ 1,

E(M, Q

, Q

, ∆, I

)  (1 + ∆M + M

Q

) (4.6)

× E

(M, Q

, Q

, ∆M + M

Q

, I

).

P r o o f. Obviously it is sufficient to prove

(4.7) m = v(q)m + ψ e

+ O

(∆M + M

Q

).

We observe that

 t(m, q

, q

) 4α(α − 1)



= (q

q

)

mσ(m

, q

, q

).

The inequality in (4.1) is equivalent to

|t

(m, q

, q

) − t

( e m, e q

, e q

)|  ∆T

, which implies

|v(q)mσ(m

, q

, q

) − e mσ( e m

, e q

, e q

)|  ∆M.

Applying Lemma 2.2, we can get for every J ≥ 0, (4.8)

v(q)m + X

{v(q)σ

(q

, q

)m

− σ

(e q

, e q

) e m

} − e m   

 ∆M + M

Q

. For the choice of J = 0, one has e m = v(q)m + O(∆M + M

Q

). Taking J = 1 in (4.8) and replacing e m by v(q)m + O(∆M + M

Q

), we get the first approximation

(4.9) m = v(q)m + ψ e

+ O(∆M + M

Q

),

namely, (4.7) holds for J = 1. Now we suppose that (4.7) is true for J − 1.

Thus we can replace e m

by {v(q)m +ψ

+O(∆M +M

Q

)}

in (4.8); we then easily deduce that (4.7) is also true for J. This finishes the proof of Lemma 4.1.

Now we divide the set Q into two sets Q

and Q

. All q satisfying (4.10) |v

(q)q

− e q

| ≥ 2|v

(q)q

− e q

|, q ∈ Q,

form the set Q

. All other q form Q

.

Lemma 4.2. Let 1 ≤ Q

≤ Q

≤ ε

M . Then for t ∼ M and q ∈ Q

, we have

(4.11) ∂

ψ

(t, q)

∂t

 |ω(q) − 1|Q

M

(i = 0, 1, 2), where

ω(q) =

 q

q

e q

e q



. P r o o f. Since

ψ

(t, q) = C{(v

(q)q

− e q

) + (v

(q)q

− e q

)} v

(q) t , by (4.10) we get

ψ

(t, q)  |v

(q)q

− e q

|M

 Q

 q

q

e q

q e



q

− e q

   M

 Q

|ω(q) − 1|M

.

Now, (4.11) is true for J = 1. Suppose that it holds for J − 1. We write ψ

= v(q)m

X

 σ

(q

, q

)

m

− σ

(e q

, e q

) (v(q)m + ψ

)

 (4.12)

− ψ

X

σ

(e q

, e q

) (v(q)m + ψ

)

.

Using (2.4), the induction hypothesis and Q

≤ Q

≤ ε

M , it is easy to see that the last term in (4.12) is

 (|ω(q) − 1|Q

M

)

 Q

M



 ε

|ω(q) − 1|Q

M

.

Similarly by (2.4) we get again

(4.13) X

σ

(q

, q

)

m

− σ

(e q

, e q

) (v(q)m + ψ

)

= C

 q

+ q

m

− q e

+ e q

(v(q)m + ψ

)



+ X

 ( P

i=0

a

q

q

)

m

− ( P

i=0

a

q e

q e

) (v(q)m + ψ

)



= C (v

(q)q

− e q

) + (v

(q)q

− e q

) (v(q)m + ψ

)

+ C (2v(q)m + ψ

)(q

+ q

)

(v(q)m + ψ

)

+

X

X

a

 q

q

m

− q e

e q

(v(q)m + ψ

)

 . Via (4.10) we have

(4.14) (v

(q)q

− e q

) + (v

(q)q

− e q

)  Q

|ω(q) − 1|.

By the induction hypothesis we have (4.15) (2v(q)m + ψ

)(q

+ q

) ψ

m

 M Q

(|ω(q) − 1|Q

M

)

M

 ε

Q

|ω(q) − 1|.

Now we prove (4.16)

X

X

a

 q

q

m

− q e

q e

(v(q)m + ψ

)



 |ω(q) − 1|Q

M

.

For any 1 ≤ j ≤ J, by the induction hypothesis we have 1

(v(q)m)

− 1

(v(q)m + ψ

)

= (v(q)m + ψ

)

− (v(q)m)

(v(q)m)

(v(q)m + ψ

)

 jM

v(q)m+ψ

\

v(q)m

u

du  jM

M

|ψ

|

 jM

M

|ω(q) − 1| Q

M  |ω(q) − 1|

M

.

For 1 ≤ i ≤ j, we have v

(q)q

− e q

= i

v²(q)q

\

e q₁²

u

du

 iQ

|v

(q)q

− e q

|  iQ

|v

(q)q

− e q

|

 iQ

Q

|ω(q) − 1|  iQ

|ω(q) − 1|

and

v

(q)q

− e q

= i

v²(q)q

\

e q₂²

u

du

 iQ

|v

(q)q

− e q

|  iQ

|ω(q) − 1|.

So for 0 ≤ i ≤ j, we have v

(q)q

q

− e q

e q

= v

(q)q

q

− (v(q)q

)

q e

+ (v(q)q

)

q e

− e q

q e

= (v(q)q

)

(v

(q)q

− e q

) + e q

((v(q)q

)

− e q

)

 (Q

Q

+ Q

)|ω(q) − 1|

 Q

|ω(q) − 1|.

Combining the above estimates we get, for any 0 ≤ i ≤ j, q

q

m

− q e

q e

(v(q)m + ψ

)

= q

q

m

− q e

q e

m

+ e q

e q

 1

(v(q)m)

− 1

(v(q)m + ψ

)



= m

(v

(q)q

q

− e q

q e

) + e q

e q

 1

(v(q)m)

− 1

(v(q)m + ψ

)



 M

Q

|ω(q) − 1|.

Hence (4.16) follows upon summing over j ≥ 2, 0 ≤ i ≤ j.

Combining (4.12)–(4.16) we conclude that the first relation of (4.11) (namely, i = 0) holds for J. The other two can be shown similarly.

The following lemma is a key to the proof of Theorem 2. To prove the lemma, we shall use Lemma 2.1 and the idea in the proof of Theorem 1.

Lemma 4.3. If Q

Q

≤ ε

M

and 1 ≤ Q

≤ Q

≤ ε

M , then there is an

η = η(M, Q

, Q

) ∈

 max

 Q

Q

, 3L

Q

Q

 , c

c − 1



such that

(4.17) η

M

X

0≤k≤K

E

(M, Q

, Q

, δ, I

)

 M Q

Q

+ δM Q

Q

+ (M Q

Q

)

+ M

Q

Q

+ (δM

Q

Q

)

+ (δM

Q

Q

)

+ (Q

Q

)

+ (δM Q

Q

)

+ (M

Q

Q

)

.

P r o o f. We put f

(t) := v(q)t + ψ

(t, q) and use S

to denote the quantity to be estimated in Lemma 4.3. Let E

(M, Q

, Q

, δ, I

) denote the number of couples (m, q) with m ∼ M , q ∈ Q

such that (4.5) holds, and E

(M, Q

, Q

, δ, I

) be the number of couples (m, q) with m ∼ M , q ∈ Q

such that (4.5) holds. Clearly we have

(4.18) E

(M, Q

, Q

, δ, I

)

= E

(M, Q

, Q

, δ, I

) + E

(M, Q

, Q

, δ, I

).

We estimate η

P

0≤k≤K

E

(M, Q

, Q

, δ, I

) first. Define

H

:= {q ∈ Q

: |ω(q)−1| ≤ η}, H

:= {q ∈ Q

: η

/2 < |ω(q)−1| ≤ η

} with η

:= η/2

(0 ≤ l ≤ L := [log(ηQ

Q

/L)/log 2]). Noticing that the last two conditions of (4.5) imply |ω(q) − 1| ≤ η, we can write

(4.19) η

X

0≤k≤K

E

(M, Q

, Q

, δ, I

)

 η

X

q∈H⁽¹⁾_η,L+1

R(f

, δ) + η

X

0≤l≤L

X

q∈H⁽¹⁾_η,l

R(f

, δ).

For q ∈ H

, we have R(f

, δ)  M trivially, which implies that the first term on the right-hand side of (4.19) is

(4.20)  η

M {Q

Q

L + η

(Q

Q

)}  M Q

Q

Lη

in view of Lemma 3.2 in [9].

When q ∈ H

(0 ≤ l ≤ L), Lemma 4.2 shows that the function f

(t) satisfies the condition of Lemma 2.1 with µ = η

Q

/M

. Hence the second term on the right-hand side of (4.19) is

(4.21)  η

X

0≤l≤L

{δM |H

| + (η

Q

)

|H

| + |H

|}

+ η

X

0≤l≤L

(δM

Q

|H

|Lη

Q

)

+ η

X

0≤l≤L

(δ

M

Q

|H

|

η

)

+ η

X

0≤l≤L

Q

Q

(δM |H

|)

. When 0 ≤ l ≤ L, Lemma 3.2 of [9] implies that

|H

|  Q

Q

L + η

(Q

Q

)

 η

(Q

Q

)

,

so we replace |H

| by the estimate η

(Q

Q

)

in (4.21). Combining (4.19) and (4.20), a simple calculation shows that

(4.22) η

X

0≤k≤K

E

(M, Q

, Q

, δ, I

)

 M Q

Q

Lη

+ δM Q

Q

+ Q

Q

+ Q

Q

η

+ (δM

Q

Q

)

L

η

+ (δ

M

Q

Q

)

η

+ (δM Q

Q

)

η

.

Now we estimate η

P

0≤k≤K

E

(M, Q

, Q

, δ, I

) by the technique in the proof of Theorem 1. Let E

(M, Q

, Q

, δ) be the number of couples (m, q) with m ∼ M , q ∈ Q

such that the first condition of (4.5) holds. We can obtain

X

0≤k≤K

E

(M, Q

, Q

, δ, I

) ≤ E

(M, Q

, Q

, δ).

From the first condition of (4.5), we get

|v

(q)q

− e q

| < 2|v

(q)q

− e q

|.

So applying similar arguments to those for the estimate (4.11) in Lemma 4.2 for i = 0, we get

(4.23) kv(q)m + v

(q)m

k  δ + Q

Q

M

.

Let B(M, Q

, Q

, δ) denote the number of couples (m, q) with m ∼ M ,