The representation of almost all numbers as sums of unlike powers

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M. B. S. L

APORTA

T. D. W

OOLEY

The representation of almost all numbers as

sums of unlike powers

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 227-240

<http://www.numdam.org/item?id=JTNB_2001__13_1_227_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

_{representation}

of almost all numbers

as sums

of unlike

_powers

par M. B. S.

LAPORTA1

et T. D.

WOOLEY2

RÉSUMÉ. Nous prouvons dans cet article que presque tout entier

s’écrit comme la somme d’un

_cube,

d’un

_bicarré,

_..., et d’une

puissance

dixième.

ABSTRACT. We prove in this article that almost all

large integers

have a

_{representation}

as the sum of a

cube,

a

biquadrate,

..., and

a tenth _power.

1. Introduction

Although

the somewhat esoteric appearance of additive

problems

in-volving

sums of mixed _powers attracts a thinner audience than the more

conventional versions of

_{Waring’s problem,}

these mixed

_problems

have pro-vided useful

_specimens

for the

_testing

and

_development

of new

technology

since the earliest

_days

of the

_{Hardy-Littlewood}

method

_(see,

for

_example,

[4, 7, 15]).

Following early

investigations

of Roth

_{[8, 9],}

_particular

attention has focused on sums of

ascending

_powers. When r is a natural

number,

let

H(r)

denote the least number s such that all

_{sufficiently large}

_integers

n

are

represented

in the form

with xi

E N

₍₁

i

_s).

_Also,

let

_H+(r)

denote the

_{corresponding}

number

s, where we instead

_merely

seek to

_represent

almost all

_integers

_n, in the

sense of natural

density.

Then Roth

[8]

established that

H+ (2)

3,

a

conclusion that is

_{transparently}

best

_possible,

and

_subsequently

_(see

_[9])

provided

the _upper bound

_H(2)

50.

_Improving

on

_previous

work of

Vaughan

[13, 14],

Thanigasalam

[10,

11,

12],

and Brfdern

_{[1, 2],}

it has

recently

been shown

_by

Ford

_{[5, 6]}

that

_14,

and moreover Ford also

supplies

the bounds

_H(3)

72

_and,

for

_large

_r,

_gives

_H(r)

«

r2 log r.

Our purpose in this paper is to bound

H+(r)

for the smallest value of r as

_yet

unresolved.

Manuscrit re~u le 15 _septembre 1999.

1 Written while the author visited the University of Michigan at Ann Arbor, and _enjoyed the benefits of a _Fellowship from the _Consiglio Nazionale delle Ricerche _(Italy).

Theorem 1. One has

_H+(3)

8.

The lower bound

_{H+(3) ~}

5 is immediate from the observation that

+ 1 -I- 5 ~- 6

1,

and so the conclusion of Theorem 1 is not

_{astronomically}

far from the truth.

_Cursory

_computations

indicate that the methods of this _paper

_yield

a bound at least as

_strong

as

H+(4) ~

20. We note also that when r is

large,

the methods of

[6, §5]

are

_{easily adapted}

to

_give

H+(r)

«r3/2(logr)1/2.

We establish Theorem 1

_by

means of the

Hardy-Littlewood

_{method, exploiting}

recent new estimates for mean values of

smooth

_Weyl

sums

(see,

in

particular,

[3,

_{18, 21, 22, 24,}

26].

These methods

yield

a lower bound for the number of

_{representations}

in the

_prescribed

form of the

_expected

size

_{predicted by}

a formal

_application

of the circle method.

When n is a natural

_number,

let

_v(n)

denote the number of

_{representations}

of n in the

shape

with Then we obtain the

following

theorem.

Theorem 2. There is a

_positive

number T

satisfying

the

_property

that,

for

all but

_of

the natural numbers n with

₁

_n

_N,

one has

/ B 1081

v(n) »

n 2520 .

We offer an outline of the

_proof

of Theorem 2 in

_§2

_below,

wherein we

also

_negotiate

certain

_{preliminaries.}

_Plainly,

Theorem 1 is an immediate

consequence of Theorem 2.

Throughout, é

will denote a

sufficiently

small

positive number,

and will denote a

_{positive integer,}

usually

in the _range

3 ~

10. We use « and » to denote

_{Vinogradov’s}

well-known

_notation,

with

_implicit

constants

_depending

at most on E and

k,

unless otherwise

indicated. In an effort to

simplify

our

_analysis,

we

adopt

the

_following

convention

_concerning

the number 6’. Whenever - _{appears in} a

_statement,

either

_implicitly

or

explicitly,

we assert that for each E &#x3E;

0,

the statement

holds for

_{sufficiently large}

values of the main

_parameter.

Note that the "value" of E may

consequently change

from statement to

_statement,

and hence also the

_dependence

of

_implicit

constants on e.

_{Finally, when y}

is a

real number we write

~y~

for the

_greatest

_integer

not

_exceeding

_y.

The first author is

_grateful

to the

_Department

of Mathematics at the

Uni-versity

of

_Michigan,

Ann

_Arbor,

for _{its generous}

_hospitality

and excellent

working

conditions

_during

the

_period

in which this _paper was written.

2. Preliminaries to the main

_argument

In this section we describe the

_strategy

underlying

our

application

of

the circle

_method,

and at the same time record

auxiliary

estimates for later use. As mentioned in the

introduction,

we make fundamental use of

smooth

_Weyl

sums. In this

_context,

when X and Y are real numbers with

2 Y

X,

define the set of Y-smooth numbers _{up to} X

_by

:

p~n

and _p

_prime

_implies

that

_p

K}.

As

_usual,

we write

e(z)

for

e27riz,

and when k is a natural

number,

we define

Finally,

when s is a

positive

real

number,

we write

We _say that an

_exponent

is

permzssible

whenever the

_exponent

has the

property

that,

for each 6 &#x3E;

_0,

there exists a

_positive

number q

=

k)

such that whenever

_Y

_X7,

then one has

In the

_argument

used to establish Theorem

_2,

we make use of the

per-missible

_exponents

listed in the table below. For k =

4,

these

_exponents

follow from the table in

_§2

of

_[3].

When k =

5, 6, ~,

these

exponents

are

provided

in the

_appendix

of

_{[21] (but}

see

[22, §9]

for k = 7 and s =

36).

Finally,

when k =

8, 9,10,

these

exponents

are recorded in

_{[22, §§10,11,12],}

on

_noting

the remarks

concerning

_process DS

concluding §8

of that _paper.

Note that the

_exponent

As

in these sources

_corresponds

here to our

We take 6 =

10-1°,

and

fixq

_to be _a

positive

number,

small

_enough

_so that

for each s and k listed in the

_table,

whenever X is

_sufficiently

_large

and

Y

X7,

one has

Consider next a

positive

number N

sufficiently large

in terms of _71, and define

When n is an

_integer

with

N/2

_{n ~}

_N,

we consider the number

v* (n)

of

_{representations}

of n in the form

₍₁₎

with

_P3/2

and Xk-2 E

A(Pk,

(4

k

10).

Plainly,

one has

v(n) ~ v*(n)

for each such

_integer

n. For the sake of

_concision,

we

_modify

the notation introduced in

₍₂₎

_by

writing

Also,

we write

and when % C

_~0,1),

we define

Then

_{by orthogonality,}

one has

v*(n)

=

v*(n;

[0,1)).

We estimate the

_integral

₍₅₎

_by

means of the circle method. Our

_primary

Hardy-Littlewood

dissection is defined as follows. Write L =

(log

N)6,

and

denote

_{by q3}

the union of the

_major

arcs

with

_{0 a q}

L and

_{(a, q)}

= 1.

Also,

define p =

[0,1) B ~P.

The

object

of the first

_phase

of our

analysis

is to show

_that,

for a suitable

_positive

number T,

p

whence,

as a _consequence of Bessel’s

inequality,

p

This first

_objective

we achieve in three

_steps.

Define 9R to be the union of

the arcs

Ç)/A A

with

_{0 a q}

P3 ~4

and

_(a,

_{q) =}

_1,

and write m =

~0,1 ) ~

9N. In

§3,

_we

estimate the minor arc contribution

v* (n;

m)

in mean _square. Let 9t denote

the union of the arcs

with

_{0 a q}

Nb

and

_{(a, q)}

= 1. Then in

§5

_{we prune} the _{set 9it} down

in mean _square. We _prune down to the thin

_{set q3}

in

_{§6, thereby completing}

the

_proof

of

_(7).

_Experts

will

_recognise

that the

_primary

_difficulty

in our

analysis

lies with the small number of classical

_Weyl

sums

_present

in

(5).

Thus,

while the work in

_§§3

and 5 is

_essentially

_routine,

the

_pruning

process of

_§6

_requires

a technical lemma not available in the literature.

Fortunately,

recent work of Brudern and

_Wooley

_[3]

_(see

also

_[22,

Lemma

_5.4])

_provides

the

_inspiration

to surmount the latter

_difficulty

in

_§4.

In the second

_phase

of our

analysis,

in

§7,

we

employ

_major

arc

technol-ogy familiar to aficionados of the new iterative methods in order to establish a lower bound of the

shape

uniformly

for

_N~2

_n N. In combination with

_(7),

this lower bound shows that

for all but

_{O (NL-T ~3 )}

of the

_{integers n}

with

_N/2

n N. The conclusion of Theorem 2 follows

_immediately,

whence also Theorem 1.

3. The minor arc contribution

Our

_goal

in this section is to estimate

_{v* (n; m)}

in mean _square, and this we achieve with a swift

application

of Holder’s

inequality.

We remark that

mixed mean values

_{incorporating}

efficient

_differencing

_processes of the

_type

used

_by

Ford

_[6]

are not worthwhile in the

_present

context.

_Although

we

have

_developed

more efficient _processes that do

_improve

the

quality

of our

bounds

_here,

it

_transpires

that such

_improvements

leave no visible trace in our final

_analysis.

Define the

_numbers

_{(4 fi k fi 9)}

as in the table

above,

and define slo

by

means of

s41

+ .- . + 1. Then

_recalling

₍₄₎

and

_applying

Holder’s

inequality,

we obtain

Suppose

that

5 fi k fi

10. Then on

_{writing tk =}

[Sk

+

_1]

_{and Ok}

= tk -

Sk,

Also,

in view of the definition of m, it follows from

[16,

Lemma

1]

that

Thus,

if we write i and

suppose that is a

permissible

_exponent,

then we obtain

On

_recalling

₍₃₎

and the table of

_exponents

from

_{§2, therefore,}

a modicum

of

_computation

reveals that with a real number

_{ø exceeding}

_0.0023,

4.

_Preparations

for

_pruning

Before

_initiating

the first

_pruning

_process, we record some notation and recall certain

_auxiliary

estimates. Write

Also,

when k §

2,

define

and define the

_{multiplicative}

function on

_prime

_powers

7r’

by taking

Then

_by

_[17,

Lemma

_3],

whenever a E Z and _q E N

_satisfy

_{(a, q)}

=

1,

_one

has

Next define for a E

_~0,1)

_{by taking}

when a E

_{VJ1( q, a)}

C

_9H,

and

_{by taking}

this function to be zero otherwise.

Then

_by

_[19,

Theorem

_4.1],

- ,,., .

We note also that

_{by applying partial integration}

to

_(11),

it follows from

(13)

and

₍₁₄₎

that whenever a E

_{m(q, a)}

C and

_(a,

_q)

=

Before

_describing

our technical

_{pruning lemma,}

we define an

_auxiliary

set of

_major

arcs. When

1 X

P3,

let denote the union of the

intervals

with

Lemma 1.

_Suppose

that

k ~

4,

1

X

Pk

and A &#x3E; 1. Write t =

[k/2],

and take A to be a subset

_of

fl7G.

_Define

the

_function

T(a)

_for

a E

_{by taking}

when

_).

Then

_for

each e &#x3E;

_0,

Proof.

We follow

_closely

the

_argument

of the

_proof

of

_[22,

Lemma

_5.4],

noting

initially

that the

_argument

_leading

to

_inequality

_(5.8)

of that _paper shows that

where

_{Q(q) _}

_¿rlq

_rK,k(r)2t.

The function

_Kk(r)

is

_{multiplicative}

with

re-spect

to r, and thus is likewise a

multiplicative

function of _q.

Further,

the

_argument

of the

_proof

of

_[22,

Lemma

_5.4]

_provides

for each

_prime

num-ber p the upper bounds and

Q(ph) «ph/k

(h ~

2).

When

k ~

4, therefore,

we deduce from

(12)

that

The

_{multiplicative}

_properties

of

_Q(q)

and

_{K3 (q)}

thus ensure that for a

suit-able constant B

_depending

at most

_{on k,}

The conclusion of the lemma now follows

immediately

from

(18).

0

We also

_require

a weak estimate of

Weyl

_type

for the

_generating

function

fio (a) .

Proof. Suppose

that cx

E VJ1 B

_By

Dirichlet’s

_{approximation theorem,}

there exist a E Z and _q E N with

_{(a~) ==}

_1,

_{1 q N1-a}

and

_{qa -}

N6-1.

But since

_{Q rt}

one

_{necessarily has q}

&#x3E;

N6. Thus,

on

_applying

[25,

Lemma

_3.1]

with k =

10,

M = and t _{= w} =

30,

_we deduce that

10

whenever _p60 = 50 ₊ A is _a

permissible

exponent,

one has

- .

-But we find from the table in

§2

that A = 0 is

admissible,

and _so

I flo (a)

_C

The conclusion of the lemma follows

_immediately.

0 5. A wide set of

_major

arcs

In this section we _prune the

major

arcs VJ1 down to the set in prepara-tion for further

_pruning

in the next section. We

_begin

_{by replacing}

the

gen-erating

function

_F3(a),

_implicit

in

_{v*(n; VJ1),}

_by

its

_{approximation}

_{F3 (a).}

In this

context,

define

Lemma 3. One has

I

Proof.

On

_recalling

₍₄₎

and the values

_{of sk}

from

_§3,

it follows from Holder’s

inequality

that

A

_comparison

of

₍₉₎

and

₍₁₅₎

thus reveals that the

_argument

of

_{§3 leading}

to

₍₁₀₎

_again

_applies,

and the conclusion of the lemma follows. D We next

_dispose

of the contribution of the set of arcs

9A B

Lemma 4. One has

Proof. Recalling

(19)

and

_applying

Holder’s

_inequality,

a trivial estimate for

_f9(a)

_yields

-where

and here we take

_{t5 = 9, t6 = 12,}

_{t7 =}

_18,

_{t8 =} 36. On

_recalling

the

permissible

exponents

from the table in

_§2,

_moreover, one has

In order to estimate

_Ji,

we note that

_by

₍₁₃₎

and

_(16),

whenever a E

9R(q, a)

C

_9N,

one has

IF;(a)1

I

G

P30(a)1/3,

where

_0(a)

is the function defined for a E fiT

_{by taking}

_{0(a) _}

when a E

_{9R(q, a)}

C

9R. Observe also that

_If4(a)12

= where

0(l)

denotes the

number of solutions of the

_{equation zi - z2}

_{= l, with zi}

E

_A(P4,

_P2)

(i

=

1, 2).

Plainly,

one has

qb(0) «

_P4

and

f4(~)2 W

and so

_by

[2,

Lemma

2],

it follows that

---

--On

_recalling

Lemma

_2,

we therefore deduce from

(20)

and

(21)

that

- - ,

-and this suffices to establish the conclusion of the lemma. D

6.

_Pruning

By

wielding

the technical

_pruning

lemma

_prepared

in

_§4,

we are able to

prune the set of arcs 91 down to the thin set

_fl3

in a

single

stroke.

Lemma 5. One has

Proof. Suppose

that

L ~ X

N6,

and define

_{93(X) =}

_{M(2X) )}

where is as defined in

§4.

Then

[20,

Lemmata 7.2 and

8.5]

show that

the former lemma

_applying

in the interval

_{X ~}

_N6,

and the latter for

_{X ~}

_Recalling

₍₁₉₎

and

_applying

Holder’s

_inequality,

we obtain

where

But

_by

_(16),

one has C where

_T(a)

is the function defined for a E as in

(17).

Thus it follows from Lemma 1 that whenever

L ~ X ~

Nb,

one has

Jk

(k

=

4,5).

On

substituting

the

latter estimates into

_(23),

and

_making

use also of

(22),

we deduce that

In order to

_complete

the

_proof

of the

_lemma,

we have

_merely

to note

that % ) fl3

is contained in the union of the sets as X runs over the

values

21 L

with 1 &#x3E;

0 and

N6. On

_summing

over the latter values of

X,

it follows from

₍₂₄₎

that the desired conclusion does indeed hold. D

Collecting together

(10)

with the conclusions of Lemmata

_3,

4 and

_5,

we

find that

whence the desired estimate

₍₆₎

follows

_immediately.

7. The main term

Before

_establishing

the lower bound

_(8),

we introduce some further

no-tation. Write _c~ for

_p(1]-l),

where

_p(t)

is the Dickman function

_(see,

for

example,

[19, §12.1~).

For our _purposes here it suffices to note

_only

that

when 1]

&#x3E; 0 one has _cn &#x3E; 0.

_Next,

when

_10,

define

_{fi (a)}

for a E

_fl3

by taking

when a E

_q3(q,

_a)

C

_fl3.

As a _consequence of

[23,

Lemma

8.5],

one has

Lemma 6. One has

Proof.

We

_{begin by replacing}

the

_exponential

sums

F3(a)

and

fk(a)

by

their

_{approximations}

and

_{fk (a).}

Write

Then since the measure

of q3

is

O(L3N-1),

on

making

liberal use of trivial

estimates for

_generating

_functions,

one finds from

(26)

and

(27)

that

But on

_recalling

(25)

and

(14),

we have

where

(30)

and

We

_complete

the

_{singular integral}

_Jo(n)

to obtain the new

integral

On

_recalling

_(11),

a

_partial

_integration

_yields

the bounds

On

_substituting

these bounds into

₍₃₀₎

and

_(32),

we deduce that

We may rewrite

(32)

in the form

When

_N/2

_n

_N,

_therefore,

one

certainly

has that

and hence an

application

of Fourier’s

_integral

formula

_rapidly

establishes

that

Next we turn our attention to the

_singular

_series,

which we

complete

to

obtain

Recalling

first

₍₁₃₎

and

_(31),

we have

A(q, n)

W

_{q~3(q)~4(q) .. -}

and in

_particular,

_by

virtue of

_(13),

one has the _upper bound

A(q, n)

C

q-3/7.

When 7r is a

_prime

number and h = 1 _or

2,

moreover, the formulae

(12)

ensure that

A(,7rh,

_n)

W

7r"~.

But the standard

_theory

of

_exponential

sums

shows that

_{A(q, n)}

is a

_{multiplicative}

function of _q

(see,

for

_example,

_[19,

§2.6]).

Thus it follows that whenever 7r is a

_prime

number and 0

0 fi

1/35,

Consequently,

there is a fixed

positive

number B with the

_property

that

whence

Thus we arrive at the conclusion

Next write

and observe that

_by

_(35),

one has for each

prime

1r that

The

_{multiplicative}

_property

of

_{A(q, n)}

_together

with the latter estimate shows that we _may rewrite

_6(n)

as an

absolutely

_convergent

_product

_{6 (n) =}

We aim now to show that

_C~(n)

»

_1,

_uniformly

in n.

Assum-ing

this

_inequality,

it follows from

_(28),

_{(29), (33),}

₍₃₄₎

and

₍₃₆₎

that for

N/2n~N,onehas

which

_yields

the conclusion of the lemma.

In order to establish that

_{C~ (n)}

W

1,

we

_{begin by}

_noting

that the

_proof

of

_[19,

Lemma

_2.12]

shows that

when h ~

1,

one has

where

_n)

denotes the number of

_incongruent

solutions of the

congru-ence

When 7r is not

_equal

to 3 and h =

1,

it follows from the

Cauchy-Davenport

theorem

_(see

_[19,

Lemma

_2.14])

that the _congruence

₍₃₈₎

is soluble with

7r t

xl. When

-xh

=

9,

_on the other

hand,

the latter conclusion is

easily

verified

_by

hand. Then the methods of

_{[19, §2.6],}

in combination with

_(37),

therefore show that for a

sufficiently large

but fixed

_positive

number

C,

one

has

uniformly

in n. This

completes

the

proof

of the lemma. 0 In view of the discussion

_{concluding §2,}

the

_proofs

of Theorems 1 and 2 follow

_immediately

from

₍₆₎

and the conclusion of Lemma 6.

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M. B. S. LAPORTA

Dipartimento di Matematica e _Appl.

Complesso Universitario di Monte S. _Angelo Via Cinthia 80126 _Napoli Italia E-mail : _{laportaOmatna2.dma.unina.it} T. D. WOOLEY Department of Mathematics

University of Michigan, East Hall 525 East University Avenue Ann _{Arbor, Michigan} 48109-1109 U.S.A.