The value of additive forms at prime arguments

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

R

OGER

J. C

OOK

The value of additive forms at prime arguments

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 77-91

<http://www.numdam.org/item?id=JTNB_2001__13_1_77_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The value of additive

forms

at

_prime

_arguments

par ROGER J. COOK

RÉSUMÉ. Soit

_{f (p)}

une forme de

_degré

k en s variables _p1,p2,...,

ps, où les pi sont des nombres premiers. On suppose que les coef

ficients _03BBi de sont

_réels,

et

_qu’au

moins l’un des _rapports

_03BBi/03BBj

est irrationnel et

_algébrique.

Si s est assez

_grand

alors

_{f prend}

des valeurs

_proches

de tous les termes d’une suite arbitraire de

nombres

_{bien-espacés.}

Cela

_généralise

des travaux antérieurs de

Brüdern,

Cook et Perelli

_{(formes linéaires),}

et Cook et Fox

_(formes

quadratiques).

La _preuve de ce résultat

dépend

du Lemme de Hua

et,

pour k ~

6 des améliorations dues à Heath-Brown de ce lemme.

ABSTRACT. Let

_{f (p)}

be an additive form of

degree

k with s prime

variables _{p1,p2, ..., ps.}

_Suppose

that

_f

has real coefficients _03BBi with

at least one ratio

_03BBi/03BBj

_algebraic

and irrational. If s is

_large

enough

then

_f

takes values close to almost all members of _any

well-spaced

sequence. This

complements

earlier work of

Brüdern,

Cook and Perelli

_(linear

_forms)

and Cook and Fox

_(quadratic

forms).

The result is based on Hua’s Lemma _{and, for k ~ 6,}

Heath-Brown’s _improvement on Hua’s Lemma.

1. Introduction

Montgomery

and

_Vaughan

_[17]

have shown that the

_exceptional

set in

Goldbach’s

_problem

satisfies

for some A &#x3E; 0.

_Recently

Li

_{[15, 16]}

has shown that we _may take A = 0.079

and A = 0.086. If the Riemann

_Hypothesis

is true for all Dirichlet

L-functions then

₍₁₎

holds for _any A

_{.1 .}

This is a classical result due to

Hardy

and Littlewood

_[10].

Brudern,

Cook and Perelli

_[2]

obtained an

analogous

result for

_binary

linear forms with irrational coefficients. Instead of

_{counting exceptional}

integers,

suitably spaced

real _{sequences mimic} the role

_{played by}

even

in-tegers

in Goldbach’s

_problem.

An

_increasing

_sequence vl v2 ... of

positive

real numbers is called

_well-spaced

if there exist

_positive

constants

C &#x3E; c &#x3E; 0 such that

Theorem 1

_(Brüdern,

Cook and

_Perelli).

Let

_{Al, A2}

be

_positive

real

num-bers.

_Suppose

that is irrational and

_algebraic.

Let V be a

well-spaced

sequence. Let 8 &#x3E; 0. Then the number

_of

v E V with v X

_for

which

has no solution in does not exceed

0(~s+~~)~

_for

_any e &#x3E; 0.

Any well-spaced

sequence satisfies

whence the bound in the theorem is non-trivial for 8

_~.

For small values of 8 the

_exceptional

set estimate is much

_stronger

than

_(1).

One reason for

this is that the Fourier transform method for

_inequalities

has a

_single

_major

arc centred at the

_origin,

whereas the

_major

arcs in Goldbach’s

_problem

bring

in arithmetic considerations

_involving

zeros of

_L-functions,

which

have to be considered with some

uniformity

with

_respect

to the modulus. Hua

_[12]

considered the

_equation

and showed that the

_exceptional

set

_{El (X) (now}

_subject

to certain

neces-sary congruence conditions modulo

8,

3 and

₅₎

satisfies

for some constant B &#x3E; 0 and Schwarz

_[18]

showed that this estimate holds with _any constant B &#x3E; 0.

_Leung

and Liu

_[14]

have shown that

for some A &#x3E; 0 and more

recently

Bauer,

Liu and Zhan

[1]

have shown

that we _may take

~

Cook and Fox

_[3]

obtained an

analogous

result for

Diophantine inequalities,

and

_again

the result for

_inequalities

is

_stronger

than that which can at

present be obtained for

_equations.

Theorem 2

_(Cook

and

_Fox).

Let

_{Ai , A2 ,}

_A3

be

_positive

real numbers.

Sup-pose that is irrational and

algebraic.

Let V be a

well-spaced

_sequence.

has no solution in

_primers

_{Pl, P2, P3}

satisfies

for

any &#x3E; 0.

Here we shall

_generalize

these results to kth _powers, and in a

_slightly

strengthened

form when k &#x3E; 6.

Theorem 3. Let k &#x3E; 3 be an

_integer

and either

Let

_A,,

... ,

Às

be

positive

real numbers.

Suppose

that

Al /A2

is irrational

and

_algebraic.

Let V be a

well-spaced

_sequence. Let 6 &#x3E; 0. Then the number

_Eo(X)

_of

v E V with v X

for

which

has no solution in

_primes

_pi, ... , ps

satisfies

for

anye &#x3E; 0.

We may suppose that 6

31

since otherwise the result is

trivial;

in

particular

we have 6

144.

Most of the conditions in the theorem can

be relaxed. We write A =

Ai/A2.

We

only

require

that A should _not be

too

_{well-approximable.}

For

_{algebraic A}

this holds

_by

Roth’s

_theorem,

but

is also true for almost all A in the sense of

_Lebesgue

_measure; see Cook

and Fox

_[3]

for a more

general

_setting

of the results. The referee has

kindly

pointed

out the

_following

two immediate _consequences of Theorem

_3,

which enable the result to be stated without reference to the _sequence V. _{To prove}

Corollary

1 consider the cases m odd and m even

_separately

and choose a

suitable v from each

interval,

an

exceptional

value if

possible,

to

_produce

a

well-spaced

sequence.

Let

Let

_{À1’... , Às}

be

_positive

real numbers.

_Suppose

that

_{Al /A2}

is irrational and

_algebraic.

Let 6 &#x3E; 0. Then the number

_of

intervals

_(m,

m +

_1]

with an

integer

m

satisfying

0 m X and

_containing

an

exceptional

value v

for

which the

_inequality

is insoluble

_satisfies

Alternatively,

the result can be stated in terms of the measure of the set

of

_exceptional

values _v; if the measure of the set was

larger

than the bound

stated the set could not fit into the intervals

_{provided by Corollary}

1.

Corollary

2. Let l~ &#x3E; 3 be an

_integer

and either

Let

I

Let

_{À1’... , Às}

be

_positive

real numbers.

_Suppose

that is irrational and

_algebraic.

Let 6 &#x3E; 0. Then the total measure

_of

the set

_{of exceptional}

values v in the interval 0 v X is

Acknowledgement.

I am

grateful

to Trevor

_Wooley

for his

_interesting

sug-gestions

on both the content and title of this _paper.

2. Preliminaries

In this section we

begin

our

approach

to the theorem via a Fourier

trans-form method

_originating

in the work of

_Davenport

and Heilbronn

_[6].

We shall

_actually

_prove more than the theorem.

We

_regard

_{k, A, ... , A,}

and 6 as constant. We take P =

and q

as a fixed small

_positive

number. Let 0 T

_1,

we shall

eventually

take

T =

X -~.

For

any v E R it is our intention to estimate the sum

with a

_non-negative

weight

W (~)

=

WT (~)

such that

W (~)

= 0 _T.

with a

_{non-negative weight.}

The

key

idea of

Davenport

and Heilbronn was

to use the Dirichlet kernel

...

which satisfies

and

We shall follow a variant of this

method,

introduced

by Davenport

(4~.

He constructed a kernel which vanished

rapidly

using

an iterative process,

starting

with the _square-wave kernel

We shall need a kernel which is

_non-negative

so start the iterative _process

with

_Ko(a)

_and,

_following

Lemma 1 of

_Davenport

_~4~,

obtain the kernel

We shall choose n to be a

large

even

_integer

and write r for n +

_2,

to be

explicit

we can take r = 4k. Then the kernel

Kl (a)

is _a

non-negative

_even

function such that

- I , - I.. -,

Further

where is an even function such that 0

1,

0 (~)

= 0

for ~ &#x3E;

1

and

_{for ~ 9.}

_Finally

we

modify

the kernel

Kl(a)

to count the solutions of our

inequality by taking

Then

provides

the

_required

_non-negative

_weight

W with ( and

where

and for _any function

_{f (a),}

_{fj (a)}

denotes

_f(Àja).

The main contribution to the

_integral

₍₈₎

should arise from the

neigh-bourhood of the

_origin

and we divide the real line into 3

_parts

and

with Y and Z = Pil where u is less

than,

but close

_to,

-k

+ 1

and ~c will be chosen later

(a

suitable choice

is IL

= 5).

3. The

_major

arc

The main contribution to the

_integral

₍₈₎

should arise from the

_major

arc M. Our treatment of the

_major

arc is based on the methods of

_Vaughan

~19],[20~.

We

_approximate

the sum

S(a)

by

the

integral

We _{use p} =

fl

₊

i-y

to denote _a

typical

_zero of the Riemann zeta function

and

_E*

to denote summation over all those _p with

I -y 1:5

T and

B3

&#x3E; 3,

1 31

where T = P) . We then take

and

Our first Lemma is

_essentially

Lemma 8 of

_Vaughan

_[20],

which in turn

depends

on Lemma 8 of

Vaughan

[19].

There are two

_changes

that we have

made. The first is to

_weight

the sums

S(a),

and

_integral

_{I (a),}

with a

log-arithmic

_weight,

this does not affect the

_argument

_{significantly.}

The other

change

is to

_{replace Vaughan’s}

estimates

exp(-2(log P) 5 )

_by

an estimate

exp(-2(log P)e)

for

_{any 0}

_1.

This can be

justified

on

_noting

that the

[19]

and

_{depends essentially}

on the zero free

_region

for the Riemann zeta

function.

_Using

the estimate

with t = 3+

(

for _{a zero}

p =

(3

₊

iq

the

exponent 5

can be

_{replaced by}

any 8

1.

Lemma 1. Let 8 be a

_Positive

number with 8

_l.

Let Y = pu with

Then

and

On M we

replace

the term

_{81 ...8s}

in the

_{integrand by}

_{Il.. - Is .}

Replac-ing

the sums one at a time we can estimate the difference

,. ~

-using

Cauchy’s

inequality

and the

_previous

lemma.

Next we

replace

the

integral

over M

_by

an

_integral

over the real

_line,

the difference is

Lemma 2. Let

_{~2 &#x3E; ...}

1.

_{Then, uniformly for}

all 0 T 1

and all v ~

2 X,

_{X ~}

we have

This is now

straightforward, using

the methods of Lemma 51 of

Daven-port

[5]

or Lemma 5 of

_Brfdern,

Cook and Perelli

_[2],

since the conditions

we have

imposed

ensure that there is a non-trivial real solution of the

in-equalities

in some suitable s-dimensional box. We shall see later how the

conditions on v _may be removed.

_Clearly

we _may renumber the variables

to order the coefficients

_a2,

since the

_{algebraic irrationality}

of

_plays

no

_part

in the

_major

arc estimate.

4. Lemmas of Hua and Heath-Brown

In this section we extend a lemma of Heath-Brown to

Diophantine

in-equalities.

We take the

_exponential

sum

A well-known lemma of Hua

_[13]

_provides

the bound

Lemma 3

_{(Hua’s Inequality) .}

For _any E &#x3E; 0 and

_for

_any

_integer

1 in the

range 1 1

k,

we have

When k &#x3E; 6 Heath-Brown

_[11]

_sharpened

the result in the case 1 = k.

Lemma 4

_{(Heath-Brown).}

_{Let k &#x3E;}

6 and E &#x3E; 0. Then

The

_analogous

result for

_{Diophantine inequalities}

is

Lemma 5. Let k &#x3E; 6 and E &#x3E; 0. Then

_for

any

fixed

non-zero real A

Proof.

We

_{begin by}

_observing

that the lemmas of Hua and Heath-Brown bound the number of

_integer

solutions of a certain

_equation

and eo

_ipso

bound the number of solutions in

_primes.

The

_exponential

sum

S(a)

a

multiplicity

bounded

by

(log P)

and this

weighting

_may be absorbed in

the factor PE. Thus for _any m

Since the

integrand

in

₍₂₄₎

is an even

function,

_changing

the variable of

integration

and

_using

the bounds for the kernel

_K(a)

we can bound this

integral by

as

required.

D

We shall also need the

_{corresponding analogue}

of Hua’s

_Lemma,

which

may be obtained in the same _way.

5. The tail

The

_simplest

contribution to estimate is that

_coming

from the tail t. We

give

the details

_6,

the case 6 is similar. We write

_Sj

_{( 0152)}

for

S(Aja).

Lemma 6. For Z with _any

_fixed

_uc &#x3E; k6

r I .

Proof.

For

_{each j}

we have

.., "

Since the

_integrand

is an even

function,

changing

the variable of

integra-tion to

_fl

= Aa and

using

the bounds for the kernel

K(a),

_{we can} bound

The lemma now follows

_using

Holder’s

_inequality

and the trivial estimate

O

We need the contribution

₍₂₅₎

from the tail to be small

_compared

to the contribution from the

_major

_arc, that is

or

Now k 2:: 3,

k6

_{3 ~}

and r = 4k _so this will be satisfied if

it

&#x3E; ~ .

We

choose a fixed

value it

in the

interval )

&#x3E; _it

&#x3E; ~ .

For

_example,

we could

specify

it

= ~.

6. The minor arc m

Now it

_only

remains to discuss the contribution of the

_pair

of intervals

to the

_integral

_(8).

An essential

_part

is

_{played by}

an estimate of Harman

[9]

for

weighted

exponential

sums in

place

of the usual

_Weyl

estimate.

(The

estimate is an

extension to

_higher

_powers of an earlier result for

_{quadratics by}

Ghosh

[8]).

We

_replace

_S(a)

_by

the

_weighted

sum

which has a more

_powerful

estimate available than that known for

un-weighted

sums.

Lemma 7

_(Harman).

_Suppose

that a has a rational

_{approximation}

a/q

with

_{(a, q)}

= 1 and

satisfying

. ~ 1 1

Let

Then

_given

_any real number E &#x3E; 0 the

_T(a)

_satisfies

Although

Harman

_proved

this estimate for sums of the form the

estimate will also hold for sums over the _range and hence for the

difference

_{T (a).}

_{Using elementary}

estimates for the number of

_prime

_powers

q P we see that

,

and therefore the bound

₍₂₆₎

also holds for

_S(a).

Lemma 8. Let a E m and _suppose that

I rv/ B I .-1 r

with

Then there are

coprime integers

_{a, q,} with

a =,4

0,

satisfying

Proof.

Let

_Q

= then

by

Dirichlet’s Theorem there exist

coprime

integers

a, q with

1 q Q

such that

Choosing u

-~

_{-f- 3}

close to the _upper

_bound,

for a E m we have

_{a ~}

0.

Our

_hypothesis

_{I S(a)}

_&#x3E;

ensures that the term

P- 2

in Har-man’s Lemma _may be

_neglected.

The condition

_q

_Q

then

_gives

Hence

Therefore the dominant term in the bracket

_{is q-l}

and hence

Thus

1

and hence _{q «}

P 3 .

0

The next

_step

is to show that on m one of the

exponential

sums

_,S’1

and

52

is small.

Lemma 9. Let cx E m. Let

_{À1, A2}

be

_positive

real numbers such that

Al/A2

Proof.

If the Lemma is false then there is an infinite _sequence of values P for

which

_{18l(a)l, 1 S2 (a)}

are both

_{large. By}

the

_previous

lemma this

_implies

that

_{Aia, A2a}

both have

_good

rational

_{approximations.}

More

_{specifically,}

there are

_integers

_{al, a2, ql, q2} with

(ai,

qi)

=

1,

al a2 0

0,

and for i

_{= 1, 2.}

Recall that Z =

_Pl’,

_so Then Further

3

_{and A}

₂

the ratio

Thus the

_algebraic

irrational

_~

would be

_approximable

of order

_greater

than

_{2, contradicting}

Roth’s Theorem. This contradiction establishes the

lemma. D

On the minor arc m reasonable control is

_{possible only}

on _average over v.

Lemma 10. Let 0 T 1.

Suppose

that V is a set

of

real numbers such

that

_{vl - v2 I &#x3E;}

2T whenever VI, V2 E V are distinct. Let be non-zero

real numbers with

_algebraic

and irrational.

_Then,

in the notation

introduced

_above,

_for

_any E &#x3E; 0

For v E R define

Then k,

E and

_by

₍₇₎

we have

Hence the set of

_{functions k,}

with v E V is an

_{orthogonal family}

which

I-r

can be normalized on

multiplying by

a suitable constant

_multiple

of T 12 r .

Bessel’s

_{inequality yields}

for _any

_{function 0}

E

_{L2 (R) .}

We take

to deduce that the left hand side of the

_{inequality proposed}

in the lemma

We estimate this

_{integral by}

_separating

m into two subsets.

_{Let X}

be chosen to

_satisfy

_{1 - i}

_x 1 and take

Then,

by

Lemma

_5,

on

_{Choosing x}

close to 1 -

_i.

By

symmetry

the same bound holds for m2 in

place

of ml, and hence

for m itself. D

7. Conclusion

It is now _{easy to} deduce the

following

result which

implies

Theorem 3.

Theorem 4. Let k be a

_{positive integer}

and let s be chosen as in

Theo-rem 3. Let 0 7 1.

Suppose

that

V (X ) is a

set

of real

numbers contained

in

_{2 X, X ~}

such that

_{vl -}

V2

&#x3E;

2T whenever are distinct. Let

... , 1 be real numbers and

Ai/A2

be

algebraic

and irrational. Then

the number

_{of v}

E V

_for

_which

-t- ... -f-

!~

7 has no solution

Proof.

We use the notation from the earlier sections.

Suppose

that

Nv

= 0

for some v E V. Then we must have

Hence

and this is in turn bounded

_by

To deduce Theorem 3 we _{may suppose} that 6

31

for otherwise the

estimate in Theorem 3 is trivial. Take T =

X-’6

in Theorem 2. Then

The condition 1 is

_easily

removed. We _may assume ... &#x3E;

_As

_by

renumbering

the variables. If

_A,

1 consider the

_inequality

and note that the numbers

_v/as

are still

well-spaced.

To obtain Theorem

2,

we now

_replace

X

_by

2-l X

and sum over 1 l «

_log

X .

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[21] G.L. _WATSON, On _{indefinite quadratic forms} in _five variables. Proc. London Math. Soc. ₍₃₎

3 _{(1953), 170-181.}

Roger J. COOK

Department of Pure Mathematics

University of Sheffield

Hicks Building, Hounsfield Road Shefheld S3 7RH

England