Analytic Number Theory A Tribute to Gauss and Dirichlet Part 4 pdf

Browning – “Counting rational points on del Pezzo surfaces of degree five”, in Proceedings of the Session in Analytic Number Theory and Diophantine Equations Bonn, Bonner Math.. Browning

But now (18) implies that Y14  B 1/2 /(Y11/2 Y041/2 Y24Y341/2), and (20) and (21)

together imply that Y03  Y33Y34/Y04 We therefore deduce that



Y1,Yi3,Yi4

(20) holds

Y03,Y04,Y33

Y1,Y23,Y24,Y34

Y033/4 Y043/4 Y231/2 Y241/2 Y331/4 Y341/4

Y1,Y04,Y33

Y23,Y24,Y34

Y231/2 Y241/2 Y33Y34.

Finally it follows from (17) and (21) that Y33  B 1/2 /(Y231/2 Y241/2 Y34), whence



Y1,Yi3,Yi4

(20) holds

Y04,Y13,Y14,Y23,Y34

1 B(log B)5,

which is satisfactory for the theorem

Next we suppose that (22) holds, so that (23) also holds In this case it follows

from (19), together with the inequality Y1 Y13Y14 Y03Y04, that

Y13 min

Y041/2 Y14Y24Y341/2

Y031/2 Y23Y331/2

, Y03Y04

Y1Y14  Y

1/4

03 Y043/4 Y241/2 Y341/4

Y11/2 Y231/2 Y331/4

.

On combining this with the inequality Y14  B 1/2 /(Y11/2 Y041/2 Y24Y341/2), that follows from (18), we may therefore deduce from (25) that



Y1,Y i3 ,Y i4

(22) holds

Y1,Y i3 ,Y i4

(22) holds

Y1Y13Y14Y23Y24Y33Y34

Y1,Y03,Y04,Y33

Y14,Y23,Y24,Y34

Y11/2 Y031/4 Y043/4 Y14Y231/2 Y243/2 Y333/4 Y345/4

Y1,Y03,Y04

Y23,Y24,Y33,Y34

Y031/4 Y041/4 Y231/2 Y241/2 Y333/4 Y343/4

Now it follows from (23) that Y33  Y03Y04/Y34 We may therefore combine this with the first inequality in (17) to conclude that



Y1,Y i3 ,Y i4

(22) holds

Y1,Y03,Y04

Y23,Y24,Y34

Y03Y04Y231/2 Y241/2  B(log B)5,

which is also satisfactory for the theorem

Finally we suppose that (24) holds On combining (19) with the fact that

Y33Y34 Y03Y04, we obtain

Y33 min Y04Y142Y242Y34

Y03Y2

13Y2 23

, Y03Y04

Y34

 Y04Y14Y24

Y13Y23 .

Summing (25) over Y33 first, with min{Y03Y04, Y33Y34}
Y 1/2

03 Y041/2 Y331/2 Y341/2, we therefore obtain



Y1,Y i3 ,Y i4

(24) holds

Y1,Y03,Y04,Y13

Y14,Y23,Y24,Y34

Y1Y031/2 Y04Y131/2 Y143/2 Y231/2 Y243/2 Y341/2

But then we may sum over Y03, Y13 satisfying the inequalities in (17), and then Y1

satisfying the second inequality in (18), in order to conclude that



Y1,Y i3 ,Y i4

(24) holds

Y1,Y04,Y13

Y14,Y23,Y24,Y34

Y1Y041/2 Y131/2 Y143/2 Y231/4 Y245/4 Y341/2

Y1,Y04,Y14

Y23,Y24,Y34

Y11/2 Y041/2 Y14Y24Y341/2  B(log B)5.

This too is satisfactory for Theorem 3, and thereby completes its proof

4 Open problems

We close this survey article with a list of five open problems relating to Manin’s conjecture for del Pezzo surfaces In order to encourage activity we have deliberately selected an array of very concrete problems

(i) Establish (3) for a non-singular del Pezzo surface of degree 4.

The surface x0 x1− x2x3= x2+ x2+ x2− x2− 2x2= 0 has Picard group

of rank 5

(ii) Establish (3) for more singular cubic surfaces.

Can one establish the Manin conjecture for a split singular cubic surface whose universal torsor has more than one equation? The Cayley cubic surface (8) is such a surface

(iii) Break the 4/3-barrier for a non-singular cubic surface.

We have yet to prove an upper bound of the shape N U,H (B) = O S (B θ),

with θ < 4/3, for a single non-singular cubic surface S ⊂ P3 This seems

to be hardest when the surface doesn’t have a conic bundle structure over

Q The surface x0 x1(x0+ x1) = x2x3(x2+ x3) admits such a structure;

can one break the 4/3-barrier for this example?

(iv) Establish the lower bound N U,H (B) B(log B)3 for the Fermat cubic.

The Fermat cubic x3+ x3= x3+ x3 has Picard group of rank 4

(v) Better bounds for del Pezzo surfaces of degree 2.

Non-singular del Pezzo surfaces of degree 2 take the shape

t2= F (x0 , x1, x2),

for a non-singular quartic form F Let N (F ; B) denote the number of integers t, x0, x1, x2 such that t2 = F (x) and |x|
B Can one prove

that we always have N (F ; B) = O ε,F (B 2+ε)? Such an estimate would be

essentially best possible, as consideration of the form F0(x) = x4+ x4−x4

shows The best result in this direction is due to Broberg [Bro03a], who

has established the weaker bound N (F ; B) = O ε,F (B 9/4+ε) For certain

quartic forms, such as F1(x) = x4+ x4+ x4, the Manin conjecture implies

that one ought to be able to replace the exponent 2 + ε by 1 + ε Can one prove that N (F1; B) = O(B θ ) for some θ < 2?

Acknowledgements The author is extremely grateful to Professors de la Bret`eche and Salberger, who have both made several useful comments about an earlier version of this paper It is also a pleasure to thank the anonymous referee for his careful reading of the manuscript

[BM90] V V Batyrev & Y I Manin – “Sur le nombre des points rationnels de hauteur born´ e

des vari´ et´ es alg´ebriques”, Math Ann 286 (1990), no 1-3, p 27–43.

[Bro03a] N Broberg – “Rational points on finite covers of P 1 and P 2”, J Number Theory 101

(2003), no 1, p 195–207.

[Bro03b] T D Browning – “Counting rational points on del Pezzo surfaces of degree five”,

in Proceedings of the Session in Analytic Number Theory and Diophantine Equations

(Bonn), Bonner Math Schriften, vol 360, Univ Bonn, 2003, 22 pages.

[Bro06] , “The density of rational points on a certain singular cubic surface”, J Number

Theory 119 (2006), p 242–283.

[BT98a] V V Batyrev& Y Tschinkel – “Manin’s conjecture for toric varieties”, J Algebraic

Geom 7 (1998), no 1, p 15–53.

[BT98b] , “Tamagawa numbers of polarized algebraic varieties”, Ast´ erisque (1998),

no 251, p 299–340, Nombre et r´ epartition de points de hauteur born´ ee (Paris, 1996) [BW79] J W Bruce& C T C Wall – “On the classification of cubic surfaces”, J London

Math Soc (2) 19 (1979), no 2, p 245–256.

[Cay69] A Cayley– “A memoir on cubic surfaces”, Phil Trans Roy Soc 159 (1869), p 231–

326.

[CLT02] A Chambert-Loir & Y Tschinkel – “On the distribution of points of bounded height

on equivariant compactifications of vector groups”, Invent Math 148 (2002), no 2,

p 421–452.

[CT88] D F Coray& M A Tsfasman – “Arithmetic on singular Del Pezzo surfaces”, Proc.

London Math Soc (3) 57 (1988), no 1, p 25–87.

[CTS76] J.-L Colliot-Th´ el` ene & J.-J Sansuc – “Torseurs sous des groupes de type

multipli-catif; applications ` a l’´ etude des points rationnels de certaines vari´ et´ es alg´ebriques”, C.

R Acad Sci Paris S´ er A-B 282 (1976), no 18, p Aii, A1113–A1116.

[CTS87] , “La descente sur les vari´ et´es rationnelles II”, Duke Math J 54 (1987), no 2,

p 375–492.

[Dav05] H Davenport – Analytic methods for Diophantine equations and Diophantine

in-equalities, second ed., Cambridge Mathematical Library, Cambridge University Press,

Cambridge, 2005, With a foreword by R C Vaughan, D R Heath-Brown and D E Freeman, Edited and prepared for publication by T D Browning.

[dlB98] R de la Bret` eche – “Sur le nombre de points de hauteur born´ ee d’une certaine

surface cubique singuli`ere”, Ast´ erisque (1998), no 251, p 51–77, Nombre et r´epartition

de points de hauteur born´ ee (Paris, 1996).

[dlB01] , “Compter des points d’une vari´ et´e torique”, J Number Theory 87 (2001),

no 2, p 315–331.

[dlB02] , “Nombre de points de hauteur born´ ee sur les surfaces de del Pezzo de degr´ e

5”, Duke Math J 113 (2002), no 3, p 421–464.

[dlBBa] R de la Bret` eche & T Browning – “On Manin’s conjecture for singular del Pezzo

surfaces of degree four, I”, Michigan Math J., to appear.

[dlBBb] , “On Manin’s conjecture for singular del Pezzo surfaces of degree four, II”,

Math Proc Camb Phil Soc., to appear.

[dlBBD] R de la Bret` eche, T D Browning & U Derenthal – “On Manin’s conjecture for

a certain singular cubic surface”, Ann Sci ´ Ecole Norm Sup., to appear.

[dlBF04] R de la Bret` eche & ´ E Fouvry – “L’´ eclat´ e du plan projectif en quatre points dont

deux conjugu´es”, J Reine Angew Math 576 (2004), p 63–122.

[dlBSD] R de la Bret` eche & P Swinnerton-Dyer – “Fonction zˆ eta des hauteurs associ´ ee ` a

une certaine surface cubique”, submitted, 2005.

[FMT89] J Franke, Y I Manin & Y Tschinkel – “Rational points of bounded height on

Fano varieties”, Invent Math 95 (1989), no 2, p 421–435.

[Fou98] E Fouvry´ – “Sur la hauteur des points d’une certaine surface cubique singuli` ere”,

Ast´ erisque (1998), no 251, p 31–49, Nombre et r´epartition de points de hauteur born´ ee (Paris, 1996).

[HB97] D R Heath-Brown– “The density of rational points on cubic surfaces”, Acta Arith.

79 (1997), no 1, p 17–30.

[HB02] , “The density of rational points on curves and surfaces”, Ann of Math (2) 155

(2002), no 2, p 553–595.

[HB03] , “The density of rational points on Cayley’s cubic surface”, in Proceedings of

the Session in Analytic Number Theory and Diophantine Equations (Bonn), Bonner

Math Schriften, vol 360, Univ Bonn, 2003, 33 pages.

[HBM99] D R Heath-Brown & B Z Moroz – “The density of rational points on the cubic

surface X3= X1 X2X3”, Math Proc Cambridge Philos Soc 125 (1999), no 3, p 385– 395.

[HP52] W V D Hodge& D Pedoe – Methods of algebraic geometry Vol II Book III:

Gen-eral theory of algebraic varieties in projective space Book IV: Quadrics and Grassmann varieties, Cambridge University Press, 1952.

[HT04] B Hassett& Y Tschinkel – “Universal torsors and Cox rings”, in Arithmetic of

higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr Math., vol 226,

Birkh¨ auser Boston, Boston, MA, 2004, p 149–173.

[Lip69] J Lipman – “Rational singularities, with applications to algebraic surfaces and unique

factorization”, Inst Hautes ´ Etudes Sci Publ Math (1969), no 36, p 195–279.

[Man86] Y I Manin– Cubic forms, second ed., North-Holland Mathematical Library, vol 4,

North-Holland Publishing Co., Amsterdam, 1986.

[Pey95] E Peyre – “Hauteurs et mesures de Tamagawa sur les vari´ et´es de Fano”, Duke Math.

J 79 (1995), no 1, p 101–218.

[Pey04] , “Counting points on varieties using universal torsors”, in Arithmetic of

higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr Math., vol 226,

Birkh¨ auser Boston, Boston, MA, 2004, p 61–81.

[PT01a] E Peyre & Y Tschinkel – “Tamagawa numbers of diagonal cubic surfaces, numerical

evidence”, Math Comp 70 (2001), no 233, p 367–387.

[PT01b] , “Tamagawa numbers of diagonal cubic surfaces of higher rank”, in Rational

points on algebraic varieties, Progr Math., vol 199, Birkh¨auser, Basel, 2001, p 275– 305.

[Sal98] P Salberger – “Tamagawa measures on universal torsors and points of bounded height

on Fano varieties”, Ast´ erisque (1998), no 251, p 91–258, Nombre et r´epartition de points de hauteur born´ ee (Paris, 1996).

[Sch64] L Schl¨ afli– “On the distribution of surfaces of the third order into species”, Phil.

Trans Roy Soc 153 (1864), p 193–247.

[SD04] P Swinnerton-Dyer– “Diophantine equations: progress and problems”, in Arithmetic

of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr Math., vol 226,

Birkh¨ auser Boston, Boston, MA, 2004, p 3–35.

[SD05] , “Counting points on cubic surfaces II”, in Geometric methods in algebra and

number theory, Progr Math., vol 235, Birkh¨auser Boston, Boston, MA, 2005, p 303– 309.

[Sko93] A N Skorobogatov– “On a theorem of Enriques-Swinnerton-Dyer”, Ann Fac Sci.

Toulouse Math (6) 2 (1993), no 3, p 429–440.

[SSD98] J B Slater & P Swinnerton-Dyer – “Counting points on cubic surfaces I”,

Ast´ erisque (1998), no 251, p 1–12, Nombre et r´epartition de points de hauteur born´ ee (Paris, 1996).

[Tsc02] Y Tschinkel– “Lectures on height zeta functions of toric varieties”, in Geometry of

toric varieties, S´emin Congr., vol 6, Soc Math France, Paris, 2002, p 227–247 [Tsc03] , “Fujita’s program and rational points”, in Higher dimensional varieties and

rational points (Budapest, 2001), Bolyai Soc Math Stud., vol 12, Springer, Berlin,

2003, p 283–310.

[Wal80] C T C Wall– “The first canonical stratum”, J London Math Soc (2) 21 (1980),

no 3, p 419–433.

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK

E-mail address: t.d.browning@bristol.ac.uk

Volume 7, 2007

The density of integral solutions for pairs of diagonal cubic equations

J¨ org Br¨ udern and Trevor D Wooley

Abstract We investigate the number of integral solutions possessed by a

pair of diagonal cubic equations in a large box Provided that the number

of variables in the system is at least thirteen, and in addition the number of

variables in any non-trivial linear combination of the underlying forms is at

least seven, we obtain a lower bound for the order of magnitude of the number

of integral solutions consistent with the product of local densities associated

with the system.

1 Introduction

This paper is concerned with the solubility in integers of the equations (1.1) a1x31+ a2 x32+ + a s x3s = b1 x31+ b2 x32+ + b s x3s = 0,

where (a i , b i)∈ Z2\{0} are fixed coefficients It is natural to enquire to what extent

the Hasse principle holds for such systems of equations Cook [C85], refining earlier work of Davenport and Lewis [DL66], has analysed the local solubility problem

with great care He showed that when s ≥ 13 and p is a prime number with p = 7,

then the system (1.1) necessarily possesses a non-trivial solution in Qp Here, by

non-trivial solution, we mean any solution that differs from the obvious one in

which x j = 0 for 1≤ j ≤ s No such conclusion can be valid for s ≤ 12, for there

may then be local obstructions for any given set of primes p with p ≡ 1 (mod 3);

see [BW06] for an example that illuminates this observation The 7-adic case,

moreover, is decidedly different For s ≤ 15 there may be 7-adic obstructions to

the solubility of the system (1.1), and so it is only when s ≥ 16 that the existence

of non-trivial solutions in Q7 is assured This much was known to Davenport and

Lewis [DL66].

Were the Hasse principle to hold for systems of the shape (1.1), then in view

of the above discussion concerning the local solubility problem, the existence of

2000 Mathematics Subject Classification Primary 11D72, Secondary 11L07, 11E76, 11P55.

Key words and phrases Diophantine equations, exponential sums, Hardy-Littlewood

method.

First author supported in part by NSF grant DMS-010440 The authors are grateful to the Max Planck Institut in Bonn for its generous hospitality during the period in which this paper was conceived.

c

2007 J¨org Br¨udern and Trevor D Wooley

integer solutions to the equations (1.1) would be decided in Q7 alone whenever

s ≥ 13 Under the more stringent hypothesis s ≥ 14, this was confirmed by the

first author [B90], building upon the efforts of Davenport and Lewis [DL66], Cook [C72], Vaughan [V77] and Baker and Br¨ udern [BB88] spanning an interval of more than twenty years In a recent collaboration [BW06] we have been able to

add the elusive case s = 13, and may therefore enunciate the following conclusion.

Theorem1 Suppose that s ≥ 13 Then for any choice of coefficients (a j , b j)∈

Z2\{0} (1 ≤ j ≤ s), the simultaneous equations (1.1) possess a non-trivial solution

in rational integers if and only if they admit a non-trivial solution inQ7.

Now let N s (P ) denote the number of solutions of the system (1.1) in rational integers x1 , , x s satisfying the condition|x j | ≤ P (1 ≤ j ≤ s) When s is large,

a na¨ıve application of the philosophy underlying the circle method suggests that

N s (P ) should be of order P s −6 in size, but in certain cases this may be false even

in the absence of local obstructions This phenomenon is explained by the failure of

the Hasse principle for certain diagonal cubic forms in four variables When s ≥ 10

and b1, , b s ∈ Z \ {0}, for example, the simultaneous equations

(1.2) 5x31+ 9x32+ 10x33+ 12x34= b1x31+ b2x32+ + b s x3s= 0

have non-trivial (and non-singular) solutions in every p-adic fieldQpas well as inR,

yet all solutions in rational integers must satisfy the condition x i= 0 (1≤ i ≤ 4).

The latter must hold, in fact, independently of the number of variables For such examples, therefore, one has N s (P ) = o(P s −6 ) when s ≥ 9, whilst for s ≥ 12 one

may show thatN s (P ) is of order P s −7 For more details, we refer the reader to the

discussion surrounding equation (1.2) of [BW06] This example also shows that

weak approximation may fail for the system (1.1), even when s is large.

In order to measure the extent to which a system (1.1) may resemble the

pathological example (1.2), we introduce the number q0, which we define by

(c,d) ∈Z2\{0}card{1 ≤ j ≤ s : ca j + db j = 0}.

This important invariant of the system (1.1) has the property that as q0 becomes larger, the counting functionN s (P ) behaves more tamely Note that in the example (1.2) discussed above one has q0= 4 whenever s ≥ 8.

Theorem 2 Suppose that s ≥ 13, and that (a j , b j) ∈ Z2\ {0} (1 ≤ j ≤ s)

satisfy the condition that the system (1.1) admits a non-trivial solution inQ7 Then whenever q0≥ 7, one has N s (P )  P s −6 .

The conclusion of Theorem 2 was obtained in our recent paper [BW06] for

all cases wherein q0 ≥ s − 5 This much suffices to establish Theorem 1; see §8 of

[BW06] for an account of this deduction Our main objective in this paper is a

detailed discussion of the cases with 7≤ q0≤ s −6 We remark that the arguments

of this paper as well as those in [BW06] extend to establish weak approximation for

the system (1.1) when s ≥ 13 and q0≥ 7 In the special cases in which s = 13 and

q0is equal to either 5 or 6, a conditional proof of weak approximation is possible by

invoking recent work of Swinnerton-Dyer [SD01], subject to the as yet unproven

finiteness of the Tate-Shafarevich group for elliptic curves over quadratic fields

Indeed, equipped with the latter conclusion, for these particular values of q0 one

may relax the condition on s beyond that addressed by Theorem 2 When s = 13

and q0 ≤ 4, on the other hand, weak approximation fails in general, as we have

already seen in the discussion accompanying the system (1.2)

The critical input into the proof of Theorem 2 is a certain arithmetic variant

of Bessel’s inequality established in [BW06] We begin in §2 by briefly sketching

the principal ideas underlying this innovation In §3 we prepare the ground for an

application of the Hardy-Littlewood method, deriving a lower bound for the major arc contribution in the problem at hand Some delicate footwork in §4 establishes

a mean value estimate that, in all circumstances save for particularly pathological situations, leads in §5 to a viable complementary minor arc estimate sufficient to

establish Theorem 2 The latter elusive situations are handled in§6 via an argument

motivated by our recent collaboration [BKW01a] with Kawada, and thereby we

complete the proof of Theorem 2 Finally, in§7, we make some remarks concerning

the extent to which our methods are applicable to systems containing fewer than

13 variables

Throughout, the letter ε will denote a sufficiently small positive number We

use  and  to denote Vinogradov’s well-known notation, implicit constants

de-pending 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 Finally, from time to time we make use of vector notation in order to save space Thus, for example, we may abbreviate (c1 , , c t) to c.

2 An arithmetic variant of Bessel’s inequality

The major innovation in our earlier paper [BW06] is an arithmetic variant of

Bessel’s inequality that sometimes provides good mean square estimates for Fourier coefficients averaged over sparse sequences Since this tool plays a crucial role also

in our current excursion, we briefly sketch the principal ideas When P and R are

real numbers with 1≤ R ≤ P , we define the set of smooth numbers A(P, R) by

A(P, R) = {n ∈ N ∩ [1, P ] : p prime and p|n ⇒ p ≤ R}.

The Fourier coefficients that are to be averaged arise in connection with the smooth

cubic Weyl sum h(α) = h(α; P, R), defined by

x ∈A(P,R)

e(αx3),

where here and later we write e(z) for exp(2πiz) The sixth moment of this sum

has played an important role in many applications in recent years, and that at hand

is no exception to the rule Write ξ = ( √

2833− 43)/41 Then as a consequence of

the work of the second author [W00], given any positive number ε, there exists a

positive number η = η(ε) with the property that whenever 1 ≤ R ≤ P η, one has (2.2)

 1 0

|h(α; P, R)|6dα  P 3+ξ+ε

We assume henceforth that whenever R appears in a statement, either implicitly

or explicitly, then 1 ≤ R ≤ P η with η a positive number sufficiently small in the

context of the upper bound (2.2)

The Fourier coefficients over which we intend to average are now defined by

 1 0

|h(α)|5e( −nα) dα.

An application of Parseval’s identity in combination with conventional circle method technology readily shows that

n ψ(n)2is of order P7 Rather than average ψ(n) in

mean square over all integers, we instead restrict to the sparse sequence consisting

of differences of two cubes, and establish the bound

1≤x,y≤P

ψ(x3− y3)2 P 6+ξ+4ε

Certain contributions to the sum on the left hand side of (2.4) are easily

es-timated By Hua’s Lemma (see Lemma 2.5 of [V97]) and a consideration of the

underlying Diophantine equations, one has

 1 0

|h(α)|4dα  P 2+ε

On applying Schwarz’s inequality to (2.3), we therefore deduce from (2.2) that

the estimate ψ(n) = O(P 5/2+ξ/2+ε ) holds uniformly in n We apply this upper bound with n = 0 in order to show that the terms with x = y contribute at most

O(P 6+ξ+2ε ) to the left hand side of (2.4) The integers x and y with 1 ≤ x, y ≤

P and |ψ(x3− y3)| ≤ P 2+ξ/2+2ε likewise contribute at most O(P 6+ξ+4ε) within the summation of (2.4) We estimate the contribution of the remaining Fourier

coefficients by dividing into dyadic intervals When T is a real number with

(2.5) P 2+ξ/2+2ε ≤ T ≤ P 5/2+ξ/2+2ε ,

defineZ(T ) to be the set of ordered pairs (x, y) ∈ N2 with

(2.6) 1≤ x, y ≤ P, x = y and T ≤ |ψ(x3− y3)| ≤ 2T,

and write Z(T ) for card( Z(T )) Then on incorporating in addition the contributions

of those terms already estimated, a familiar dissection argument now demonstrates

that there is a number T satisfying (2.5) for which

1≤x,y≤P

ψ(x3− y3)2 P 6+ξ+4ε + P ε T2Z(T ).

An upper bound for Z(T ) at this point being all that is required to complete

the proof of the estimate (2.4), we set up a mechanism for deriving such an upper bound that has its origins in work of Br¨udern, Kawada and Wooley [BKW01a]

and Wooley [W02] Let σ(n) denote the sign of the real number ψ(n) defined in

(2.3), with the convention that σ(n) = 0 when ψ(n) = 0, so that ψ(n) = σ(n) |ψ(n)|.

Then on forming the exponential sum

K T (α) =

(x,y) ∈Z(T )

σ(x3− y3)e(α(y3− x3)),

we find from (2.3) and (2.6) that

 1 0

|h(α)|5

K T (α) dα ≥ T Z(T ).

An application of Schwarz’s inequality in combination with the upper bound (2.2) therefore permits us to infer that

(2.8) T Z(T )  (P 3+ξ+ε)1/2

 1 0

|h(α)4K T (α)2|dα1/2.

Next, on applying Weyl’s differencing lemma (see, for example, Lemma 2.3 of

[V97]), one finds that for certain non-negative numbers t l , satisfying t l = O(P ε)

for 0 < |l| ≤ P3, one has

|h(α)|4 P3+ P

0< |l|≤P3

t l e(αl).

Consequently, by orthogonality,

 1

0

|h(α)4K T (α)2| dα  P3

 1 0

|K T (α) |2dα + P 1+ε K T(0)2

 P ε (P3Z(T ) + P Z(T )2).

Here we have applied the simple fact that when m is a non-zero integer, the number

of solutions of the Diophantine equation m = x3− y3 with 1≤ x, y ≤ P is at most O(P ε) Since T ≥ P 2+ξ/2+2ε , the upper bound Z(T ) = O(T −2 P 6+ξ+2ε) now follows from the relation (2.8) On substituting the latter estimate into (2.7), the desired conclusion (2.4) is now immediate

Note that in the summation on the left hand side of the estimate (2.4), one may

restrict the summation over the integers x and y to any subset of [1, P ]2 without affecting the right hand side Thus, on recalling the definition (2.3), we see that we

have proved the special case a = b = c = d = 1 of the following lemma.

Lemma 3 Let a, b, c, d denote non-zero integers Then for any subset B of

[1, P ] ∩ Z, one has

 1

0

 1

0

|h(aα)h(bβ)|5

x ∈B e((cα + dβ)x3)2

dα dβ  P 6+ξ+ε

This lemma is a restatement of Theorem 3 of [BW06] It transpires that no

great difficulty is encountered when incorporating the coefficients a, b, c, d into the

argument described above; see§3 of [BW06].

We apply Lemma 3 in the cosmetically more general formulation provided by the following lemma

Lemma 4 Suppose that c i , d i (1≤ i ≤ 3) are integers satisfying the condition

(c1d2− c2d1)(c1d3− c3d1)(c2d3− c3d2)= 0.

Write λ j = c j α + d j β (j = 1, 2, 3) Then for any subset B of [1, P ] ∩ Z, one has

 1 0

 1 0

|h(λ1)h(λ2)|5

x ∈B e(λ3x3)2

dα dβ  P 6+ξ+ε

Proof The desired conclusion follows immediately from Lemma 3 on making

a change of variable The reader may care to compare the situation here with that

occurring in the estimation of the integral J3in the proof of Theorem 4 of [BW06]

a mean value estimate that, in all circumstances save for particularly pathological situations, leads in §5 to a viable... 8

and q0 ≤ 4, on the other hand, weak approximation fails in general, as we have

already seen in the discussion accompanying...