Waring’s Problem: A Survey
R. C. Vaughan1 and T. D. Wooley2

1

The Classical Waring Problem“Omnis integer numerus vel est cubus, vel e duobus, tribus, 4, 5, 6, 7,
8, vel novem cubis compositus, est etiam quadrato-quadratus vel e duobus,
tribus, &c. usque ad novemdecim compositus, & sic deinceps.”
Waring [150, pp. 204-5].
“Every integer is a cube or the sum of two, three, ... nine cubes; every
integer is also the square of a square, or the sum of up to nineteen such;and so forth.”
Waring [152, p. 336].
It is presumed that by this, in modern notation, Waring meant that forevery k ≥ 3 there are numbers s such that every natural number is the sumof at most s k-th powers of natural numbers and that the smallest such
number g(k) satisfies g(3) = 9, g(4) = 19.
By the end of the nineteenth century, the existence of g(k) was known
for only a finite number of values of k. There is an account of this workin Dickson [48], and as far as we have been able to ascertain, by 1909 its
existence was known for k = 3, 4, 5, 6, 7, 8, 10, but not for any larger k (ofcourse, with the natural extension of the definition of g(k), Lagrange provedin 1770 that g(2) = 4). However, starting with Hilbert [69], who showedthat g(k) does indeed exist for every k, the twentieth century has seen
an almost complete solution of this problem. Let [x] denote the greatest
integer not exceeding x and write {x} for x − [x]. As the result of the workof many mathematicians we now know thatg(k) = 2k + [(3/2)k ] − 2,provided that
2k {(3/2)k } + [(3/2)k ] ≤ 2k .
If this fails, theng(k) = 2k + [(3/2)k ] + [(4/3)k ] − θ
1 Research supported by NSF grant DMS-9970632.
2 Packard Fellow, and supported in part by NSF grant DMS-9970440.

1

(1.1)

2

R. C. Vaughan and T. D. Wooley

where θ is 2 or 3 according as[(4/3)k ][(3/2)k ] + [(4/3)k ] + [(3/2)k ]equals or exceeds 2k .
The condition (1.1) is known to hold (Kubina & Wunderlich [85]) whenever k ≤ 471, 600, 000, and Mahler [91] has shown that there are at most afinite number of exceptions. To complete the proof for all k it would sufficeto know that {(3/2)k } ≤ 1−(3/4)k−1 . Beukers [3] has shown that wheneverk > 5, 000 one has {(3/2)k } ≤ 1 − ak , where a = 2−0.9 = 0.5358..., and thishas been improved slightly by Dubitskas [49] to a = 0.5769..., so long as k issufficiently large (see also Bennett [1] for associated estimates). A problem
related to the evaluation of g(k) now has an almost definitive answer. Let
gr (k) denote the smallest integer s with the property that every natural
number is the sum of at most s elements from the set {1k , rk , (r + 1)k , . . . }.
Then Bennett [2] has shown that for 4 ≤ r ≤ (k + 1)1−1/k − 1, one has
gr (k) = rk + [(1 + 1/r)k ] − 2.
By the way, before turning to the modern form of Waring’s problem, it
is perhaps worth observing that in the 1782 edition of Meditationes Algebraicæ, Waring makes an addition:“confimilia etiam affirmari possunt (exceptis excipiendis) de eodem numero
quantitatum earundem dimensionum.”
Waring [151, p. 349].
“similar laws may be affirmed (exceptis excipiendis) for the correspondingly
defined numbers of quantities of any like degree.”Waring [152, p. 336].
It would be interesting to know exactly what Waring had in mind. This,taken with some of the observations which immediately follow the remark,suggest that for more general polynomials than the k-th powers he wasaware that some kind of local conditions can play a rôle in determining
when representations occur.

2

The Modern Problem

The value of g(k) is determined by the peculiar behaviour of the first threeor four k-th powers. A much more challenging question is the value, for k ≥2, of the function G(k), the smallest number t such that every sufficiently
large number is the sum of at most t k-th powers of positive integers.
The function G(k) has only been determined for two values of k, namelyG(2) = 4, by Lagrange in 1770, and G(4) = 16, by Davenport [30]. The
bulk of what is known about G(k) has been obtained through the mediumWaring’s Problem: A Survey

3of the Hardy-Littlewood method. This has its genesis in a celebrated paperof Hardy and Ramanujan [64] devoted to the partition function. In thispaper (section 7.2) there is also a brief discussion about the representationof a natural number as the sum of a fixed number of squares of integers,
and there seems little doubt that it is the methods described therein which
inspired the later work of Hardy and Littlewood.
Our knowledge concerning the function G(k) currently leaves much to
be desired. If, instead of insisting that all sufficiently large numbers be
represented in a prescribed form, one rather asks that almost all numbers(in the sense of natural density) be thus represented, then the situation issomewhat improved. Let G1 (k) denote the smallest number u such that
almost every number n is the sum of at most u k-th powers of positive integers. The function G1 (k) has been determined in five non-trivial instances
as follows:
Davenport [29],Hardy and Littlewood [62],Vaughan [121],
Wooley [155],
Wooley [155],

G1 (3) = 4,
G1 (4) = 15,
G1 (8) = 32,
G1 (16) = 64,
G1 (32) = 128(of course, the conclusion G1 (2) = 4 is classical).

3

General Upper Bounds for G(k)The first explicit general upper bound for G(k), namely
G(k) ≤ (k − 2)2k−1 + 5,
was obtained by Hardy and Littlewood [61] (in [58] and [59], only theexistence of G(k) is stated, although it is already clear that in principle
their method gave an explicit upper bound). In Hardy and Littlewood [62]this is improved toG(k) ≤ (k − 2)2k−2 + k + 5 + [ζk ],
where
ζk =

(k − 2) log 2 − log k + log(k − 2).
log k − log(k − 1)

There has been considerable activity reducing this upper bound over theyears, and Table 1 below presents upper bounds for G(k) that were probably the best that were known at the time they appeared, at least for4

R. C. Vaughan and T. D. Wooley

Vinogradov [136], 32(k log k)2 ,Vinogradov [137][139], k 2 log 4 + (2 − log 16)k (k ≥ 3),
Vinogradov [135] [138] [140] [143], 6k log k + 3k log 6 + 4k (k ≥ 14),
Vinogradov [147], k(3 log k + 11),Tong [114], k(3 log k + 9),
Jing-Run Chen [24], k(3 log k + 5.2),
Vinogradov [148], 2k(log k+2 log log k+O(log log log k)),Vaughan [124], 2k(log k + log log k + O(1)),
Wooley [155], k(log k + log log k + O(1)).
Table 1. General upper bounds for G(k)k sufficiently large. This list is not exhaustive. In particular, there is along sequence of papers by Vinogradov between 1934 and 1947, and forfurther details we refer the reader to the Royal Society obituary of I. M.
Vinogradov (see Cassels and Vaughan [23]).
The last entry on this list has been refined further by Wooley [159], andthis provides the estimate
G(k) ≤ k(log k + log log k + 2 + O(log log k/ log k))that remains the sharpest available for larger exponents k.

4

Cubes

For small values of k there are many special variants of the Hardy-Littlewood
method that have been developed. However, in the case of cubes, until recently the best upper bounds were obtained by rather different methods
that related cubes to quadratic forms, especially sums of squares. Thus
Landau [86] had shown that G(3) ≤ 8, and this bound was reduced by
Linnik [87][88] to G(3) ≤ 7, with an alternative and simpler proof given by
Watson [153]. Only with the advent of refinements to the circle method
utilising efficient differencing did it become feasible (Vaughan [119]) to givea proof of the bound G(3) ≤ 7 via the Hardy-Littlewood method. Subsequent developments involving the use of smooth numbers (see Vaughan[124][125] and Wooley [158]) have provided a more powerful approach tothis problem that, from a practical point of view, is more direct than earlier
treatments. Complicated nonetheless, these latter proofs yield much moreWaring’s Problem: A Survey

5

information concerning Waring’s problem for cubes. We can illustrate thelatter observation with two examples which, in the absence of foreseeable
progress on the upper bound for G(3), provide the problems central tocurrent activity surrounding Waring’s problem for cubes.
When X is a large real number, denote by E(X) the number of positive
integers not exceeding X that cannot be written as a sum of four positive
integral cubes. Then the conclusion G1 (3) = 4, attributed above to Davenport [29], is an immediate consequence of the estimate E(X)  X 29/30+
established in the latter paper. Following subsequent work of Vaughan[119], Brüdern [10][12], and Wooley [158], the sharpest conclusion currently
available (see Wooley [162]) shows that E(X)  X 1−β for any positive
number β smaller than√
(422 − 6 2833)/861 = 0.119215 . . . .
It is conjectured that G(3) = 4 (see §10 below), and this would imply that
E(X)  1.
Consider next the density of integers represented as a sum of threepositive integral cubes. When X is a large real number, let N (X) denotethe number of positive integers of the latter type not exceeding X. It isconjectured that N (X)  X, and following work of Davenport [29][33],Vaughan [118][119], Ringrose [100], Vaughan [124] and Wooley [158], thesharpest currently available conclusion due to Wooley [162] establishes that
N (X)  X 1−α for any real number α exceeding√( 2833 − 43)/123 = 0.083137 . . . .
We remark that, subject to the truth of an unproved Riemann Hypothesis concerning certain Hasse-Weil L-functions, one has the conditional
estimate N (X)  X 1− due to Hooley [73][74], and Heath-Brown [67].
Unfortunately, the underlying L-functions are not yet known even to havean analytic continuation inside the critical strip.
We finish our discussion of Waring’s problem for cubes by noting that,while Dickson [47] was able to show that 23 and 239 are the only positive
integers not represented as the sum of eight cubes of natural numbers, nosuch conclusion is yet available for sums of seven or fewer cubes (but see
McCurley [92] for sums of seven cubes, and Deshouillers, Hennecart andLandreau [44] for sums of four cubes).

5

Biquadrates

Davenport’s definitive statement that G(4) = 16 is not the end of thestory for sums of fourth powers (otherwise known as biquadrates). Let

6

R. C. Vaughan and T. D. Wooley

G# (4) denote the least integer s0 such that whenever s ≥ s0 , and n ≡ r(mod 16) for some integer r with 1 ≤ r ≤ s, then n is the sum of at mosts biquadrates. Then Davenport [30] showed that G# (4) ≤ 14, and thishas been successively reduced by Vaughan [121][124] to G# (4) ≤ 12. In an
ironic twist of fate, the polynomial identity
x4 + y 4 + (x + y)4 = 2(x2 + xy + y 2 )2 ,reminiscent of identities employed in the nineteenth century, has recently
been utilised to make yet further progress. Thus, when n ≡ r (mod 16)
for some integer r with 1 ≤ r ≤ 10, Kawada and Wooley [82] have shownthat n is the sum of at most 11 biquadrates. This and allied identities havealso permitted the proof of an effective version of Davenport’s celebrated
theorem. Thus, as a consequence of work of Deshouillers, Hennecart andLandreau [45] and Deshouillers, Kawada and Wooley [46], it is now known
that all integers exceeding 13, 792 may be written as the sum of at mostsixteen biquadrates. A detailed history of Waring’s problem for biquadrates
is provided in Deshouillers, Hennecart, Kawada, Landreau and Wooley [43].

6Upper Bounds for G(k) when 5 ≤ k ≤ 20

Although we have insufficient space to permit a comprehensive accountof the historical evolution of available upper bounds for G(k) for smaller
values of k, in Table 2 we have recorded many of the key developments,
concentrating on the past twenty-five years. Each row in this table presents
the best upper bound known for G(k), for the indicated values of k, at thetime of publication of the cited work. We note that the claimed boundG(7) ≤ 52 of Sambasiva Rao [99] is based on an arithmetical error, andhence we have attributed the bound G(7) ≤ 53 parenthetically to Davenport’s methods [28] [32]. Also, it is worth remarking that the work ofVaughan and Wooley [130] [131] and [132] appeared in print in an orderreversed from its chronological development (indeed, this work was first announced in 1991). The bounds parenthetically attributed to Vaughan and
Wooley [132] follow directly from the methods therein, and were announced
on that occasion, though details (with additional refinements) appeared
only in Vaughan and Wooley [134]. Meanwhile, the bounds recorded in
Wooley [159] were an immediate consequence of the methods of Wooley[155] combined with the new estimates for smooth Weyl sums obtained in
the former work (no attempt was made therein to exploit the methods ofVaughan and Wooley [132]).

7
Waring’s Problem: A SurveykHardy & Littlewood [62]James [81]Heilbronn [68]
Estermann [52]Hua [76]
Davenport [28][32]
Narasimhamurti [96]Chen [24]
Cook [26]Vaughan [115]Thanigasalam [107]Thanigasalam [108]Thanigasalam [109]Vaughan [120][121]Vaughan [124][126]Brüdern [11]Vaughan & Wooley [129]
Wooley [155]
Wooley [159]Vaughan & Wooley [132]
Vaughan & Wooley [130]Vaughan & Wooley [131]Meng [94]Vaughan & Wooley [134]Conjectured
99

93

107
106
104
103

122
121

109 125 141 156

122 137 153 168
121 136 152 167
119 134 150 165

5 6
7
8
9
10
11 12 13 1441 87 192 425 949 2113
35
164
824
164 190 217 244 272 300 329
29 42 59 78 101 125 153 184 217 253
28
23 36 (53)

171

184
183
181

359
292

15

187 202 217 232 248

200 216
199 215 231 248 264
197 213 229 245 262

388 418 449 480 511
333 377 424 474

16

20

96
91
90
88
87
82
75

19

68
62
57

55

18

50
45
41

47

92 108 124 139 153 168 183 198 213 228 243
63
70 79 87 95 103 112 120 129 138 146
62
78 86 94 102 110 118 127 135 144(59) (67) (76) (84) (92) (100)

17

36

51

73

21 31
19 29
18
18 28
27

43
42

143
109 117 125 134 142
64 18 27 20 25

33

100
16

17 25
24

92
15

59
12

84
14

50
13

76
16
32

9

67
12
8

6

Table 2. Upper bounds for G(k) when 5 ≤ k ≤ 20

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R. C. Vaughan and T. D. Wooley

The Hardy-Littlewood Method

Practically all of the above conclusions have been obtained via the HardyLittlewood method. Here is a quick introduction. Let n be a large natural
number, and write P = n1/k andXf (α) =e(αxk )
x≤P

(here we follow the standard convention of writing e(z) for e2πiz ). Thenon writing R(n) for the number of representations of n as the sum of s kth
powers of natural numbers, it follows from orthogonality that
Z 1
R(n) =
f (α)s e(−αn)dα.
0

When α is “close” to a rational number a/q with (a, q) = 1 and q “small”,
we expect thatf (α) ∼ q −1 S(q, a)v(α − a/q),
where
S(q, a) =qXke(ar /q) and v(β) =Z Pe(βγ k )dγ.

0

r=1This relation is straightforward to establish in an interval about a/q, solong as “close” and “small” are interpreted suitably. Now put
Z
RA (n) =f (α)s e(−αn)dα.
A

For a suitable union M of such intervals centred on a/q (the major arcs),
and for s sufficiently large in terms of k, one can establish that as n → ∞,the asymptotic relation
RM (n) ∼Γ(1 + 1/k)s s/k−1nS(n)Γ(s/k)

holds, where S(n) is the singular series
S(n) =∞X

T (q; n)q=1

and
T (q; n) =

q
X

a=1
(a,q)=1

q −s S(q, a)s e(−an/q).

Waring’s Problem: A Survey

89

The Necessary Congruence Condition

For each prime p, let
U (p; n) =∞XT (pk ; n).

k=0

The function T (q; n) is multiplicative. Thus, when the singular series converges absolutely, one has
Y
S(n) =
U (p; n).
p

It is helpful to the success of the circle method that the singular series
should satisfy S(n)  1. With this observation in mind, Hardy and
Littlewood [61] defined Γ(k) to be the least integer s with the property
that, for every prime number p, there is a positive number C(p) such that
U (p; n) ≥ C(p) uniformly in n. Subsequently, Hardy and Littlewood [62]showed that indeed S(n)  1 whenever s ≥ max{Γ(k), 4}. Next let Γ0 (k)be the least number s with the property that the equation
xk1 + · · · + xks = n(8.1)

has a non-singular solution in Qp (or rather, that the corresponding congruence modulo q always has a solution with (x1 , q) = 1). Hardy and
Littlewood [63] were able to show that Γ0 (k) = Γ(k) (see Theorem 1 of theaforementioned paper). Thus one sees that the singular series reflects the
local properties of sums of k-th powers. In particular, the singular series iszero whenever the equation (8.1) fails to have a p-adic solution, for someprime p, and this reflects the trivial observation that the equation can besoluble over Z only if it is soluble everywhere locally.
Hardy and Littlewood [63] conjecture that Γ(k) → ∞ as k → ∞, but it
is not even known whether or not one has
lim inf Γ(k) ≥ 4.
k→∞

When k > 2, they showed that Γ(k) = 4k when k is a power of 2 and that
Γ(k) ≤ 2k otherwise. They also computed Γ(k) exactly when 3 ≤ k ≤ 36,
and established that Γ(k) ≥ 4 when 3 ≤ k ≤ 3000. Here they showed that
equality occurs only when k = 3, 7, 19, and possibly (but improbably)
when k = 1163, 1637, 1861, 1997, 2053. These values of k can probably
be settled by modern computing methods, and doubtless the calculations
could be carried a good deal further. As far as we are aware, nothing has
been done in this direction.

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R. C. Vaughan and T. D. Wooley

For a more detailed exposition of the Hardy-Littlewood method andthe analysis of the major arcs and the singular series, see Vaughan [128](especially Chapters 2 and 4).

9The Minor Arcs

In order to establish an asymptotic formula for R(n) it suffices to showthat Rm (n) = o(ns/k−1 ), where m = [0, 1) \ M (the minor arcs). One
needs to show that the minor arc contribution Rm (n) is smaller by a factoro(n−1 ), or equivalently o(P −k ), than the trivial estimate P s . Routinely
this is established via an inequality of the kindZs−2t Z 1s|f (α)| dα ≤ sup |f (α)||f (α)|2t dα.
mm

0

The integral on the right hand side of this inequality may be interpreted
as the number of solutions of an underlying diophantine equation, and it isfrom here that most of the savings usually come. On the other hand, nontrivial estimates for |f (α)|, when α ∈ m, may be obtained from estimates
stemming from work of Weyl [154] and Vinogradov [144] (see Vaughan [128]for more modern estimates). When successful, this leads to the relation
R(n) ∼

Γ(1 + 1/k)s s/k−1nS(n).
Γ(s/k)(9.1)

It is in finding ways of dealing with the minor arcs, or in modifying themethod so as to make the minor arcs more amenable, that most of theresearch has concentrated in the eighty years that have elapsed since thepioneering investigations of Hardy and Littlewood.

10

The Asymptotic FormulaeDefine G(k)to be the smallest natural number s0 such that whenevers ≥ s0 , the asymptotic relation (9.1) holds. Work of Hardy and Littlewood[62], described already in §3, established the general upper bounde
G(k)≤ (k − 2)2k−1 + 5.
e
Progress on upper bounds for G(k)has since been achieved on two fronts.
e
In Table 3 we present upper bounds for G(k)relevant for small values ofk.
When k is large, bounds stemming from Vinogradov’s mean value theorem provide dramatic improvements over the estimates recorded in Table 3.

Waring’s Problem: A Survey

11

Hua [75], 2k + 1,
Vaughan [119][122], 2k(k ≥ 3),

Heath-Brown [65][66], 7 · 2

k−3

k−3

Boklan [5], 7 · 2+1(k ≥ 6),(k ≥ 6).

eTable 3. Upper bounds for G(k):smaller kVinogradov [142], 183k 9 (log k + 1)2 + 1,
Vinogradov [141], 91k 8 (log k + 1)9 + 1(k ≥ 20),

2

Vinogradov [147], 10k log k,
Hua [77], 4k 2 (log k + 12 log log k + 8),
Wooley [156],2k 2 (log k + O(log log k)),Ford [53], k 2 (log k + log log k + O(1)).
eTable 4. Upper bounds for G(k):larger ke
In Table 4 we present upper bounds for G(k)of use primarily when k islarge.
We note that the methods underlying the last two bounds can beeadapted to give explicit bounds for G(k)when k is of moderate size. Thusethe method of Ford yields a bound for G(k)that is superior to the bestrecorded in Table 3 as soon as k ≥ 9, and indeed unpublished work of
Boklan and Wooley pushes this transition further to k ≥ 8.
The bounds recorded in Tables 3 and 4 are likely to be a long way fromthe truth. One might expect that G(k) = max{k + 1, Γ0 (k)}, and, with anappropriate interpretation of the asymptotic formula when S(n) = 0, thateG(k)= k + 1.

One curiosity is that when k = 3 and s = 7, it can be shown that
R(n)  n4/3 (see Vaughan [125]), yet we are unable to show that R(n) 
n4/3 . Indeed, it is currently the case that, quite generally, when s lies inethe range between the known upper bounds for G(k) and G(k),we canshow that R(n)  ns/k−1 , but not R(n)  ns/k−1 .

12

11

R. C. Vaughan and T. D. Wooley

Diminishing Ranges

In the Hardy-Littlewood method as outlined above the main problem isthat of obtaining a suitable estimate for the mean value
Z 1|f (α)|2t dα,0that is, the number of integral solutions of the equation
xk1 + · · · + xkt = y1k + · · · + ytk ,(11.1)

with 1 ≤ xj , yj ≤ P (1 ≤ j ≤ t). Available estimates can be improved
significantly if the variables are restricted, an idea already present in Hardy
and Littlewood [62]. Define
P1 = 12 P,

1−1/k

Pj = 12 Pj−1

(2 ≤ j ≤ t),and consider the equation (11.1) subject to the constraints Pj < xj , yj ≤2Pj (1 ≤ j ≤ t). Inspecting the expression |xkj − yjk | successively for
j = 1, 2, . . . when xj 6= yj , we find that only the diagonal solutions inwhich xj = yj (1 ≤ j ≤ t) occur. Thus we find that the number ofsolutions of this type is at most O(P1 . . . Pt ). This saves P1 . . . Pt over thetrivial bound, which is of order (P1 . . . Pt )2 . Now P1 . . . Pt ≈ P λ , where
λ = 1 + (1 − 1/k) + · · · + (1 − 1/k)t−1 = k − k(1 − 1/k)t .
Already when t ∼ Ck log k, for a suitable positive constant C, this exponent
is close to k. Vinogradov and Davenport have exploited and developed
this idea in a number of ways (see, for example, Davenport [27][34], andVinogradov [147][149]; see also Davenport and Erdős [35]).
There is a “p-adic” analogue of this idea, first exploited by Davenport[31], in which one considers expressions of the kindxk1 + pk2 (xk2 + pk3 (xk3 + · · · ))on each side of the equation. Here the pi denote suitably chosen prime
numbers. The analysis rests on congruences of the type x1 ≡ y1 (mod pk ).
When p > P 1/k and 1 ≤ x1 , y1 ≤ P , this congruence implies that x1 = y1 ,and so on, just as in the diminishing ranges device. This idea has the meritof returning the various k-th powers pk2 . . . pkj xkj to being in comparable
size ranges. However in each of these methods the variables in (11.1) havevarying natures and the homogeneity is essentially lost.

Waring’s Problem: A Survey

12

13

Smooth Numbers and Efficient Differences

In modern variants of the circle method as applied to Waring’s problem, starting with the work of Vaughan [124], homogeneity is restored byconsidering the number of solutions St (P, R) of the equation (11.1) withxj , yj ∈ A(P, R), where A(P, R) denotes the set of R-smooth numbers up
to P , namely
A(P, R) = {n ∈ [1, P ] ∩ Z : p|n =⇒ p ≤ R}.
In applications, one takes R to be a suitably small, but positive power of
P . The set A(P, R) has the extremely convenient property that, given any
positive number M with M ≤ P , and an element x ∈ A(P, R) with x > M ,there is always an integer m with m ∈ [M, M R] for which m|x. Moreover,this integer m can be coaxed into playing the rôle of the prime p in the padic argument mentioned above. Finally, and of great importance in what
follows, the set A(P, R) has positive density whenever R is no smaller thana positive power of P .
The objective now is to find good exponents λs with the property thatwhenever ε > 0, there exists a positive number η0 = η0 (s, k, ε) such thatwhenever R = P η with 0 < η ≤ η0 , one has
Ss (P, R)  P λs +ε .
Such exponents are established via an iterative process in which a sequence(n)(n)
of sets of exponents λ(n) = (λ1 , λ2 , . . .) is constructed by finding an ex(n+1)
in terms of the elements of λ(n) . Boundedness ispression for each λs
trivial, so there is always a convergent subsequence. In fact, our arguments
produce monotonicity, and the convergence is fairly rapid. For a more detailed introduction and motivation for the underlying ideas in using smooth
numbers in Waring’s problem, see the survey article Vaughan [127].
Beginning with the work of Wooley [155], a key element in the iterations
is the repeated use of efficient differencing, and this procedure is fully
exploited in subsequent work of Vaughan and Wooley [130] [131] [132] [134].
For each s ∈ N, we take φi = φi,s (i = 1, . . . , k) to be real numbers with0 ≤ φi ≤ 1/k. For 0 ≤ j ≤ k, we then define
Pj = 2j P,

Mj = P φj ,
ej =
H

j
Y

i=1

Hj = Pj Mj−k ,
Hi

fj =
and M

Qj = Pj (M1 . . . Mj )−1 ,
j
Y

Mi R.

i=1

Define the modified forward difference operator, ∆∗j , recursively by taking∆∗1 (f (x); h; m) = m−k (f (x + hmk ) − f (x)),

14

R. C. Vaughan and T. D. Wooley

and when j ≥ 1, by inductively defining∆∗j+1 (f (x); h1 , . . . , hj+1 ; m1 , . . . , mj+1 )= ∆∗1 (∆∗j (f (x); h1 , . . . , hj ; m1 , . . . , mj ); hj+1 ; mj+1 ).
For 0 ≤ j ≤ k, letf (z) = (z − h1 mk1 − · · · − hj mkj )k ,and define the difference polynomialΨj = Ψj (z; h1 , . . . , hj ; m1 , . . . , mj )by takingΨj = ∆∗j (f (z); 2h1 , . . . , 2hj ; m1 , . . . , mj ).
Here we adopt the convention that Ψ0 (z; h; m) = z k . We writeX
fj (α) =e(αxk ),x∈A(Qj ,R)andFj (α) =Xe(αΨj (z; h; m)),z,h,mwhere the summation is over z, h, m with1 ≤ hi ≤ 2j−i Hi ,1 ≤ z ≤ Pj ,Mi < mi ≤ Mi R,

mi ∈ A(P, R),

for 1 ≤ i ≤ j. Finally, we define
Z 1
T (j, s) =|Fj (α)fj (α)2s |dα.
0Now, on considering the underlying diophantine equation, we have
Z 1
Ss+1 (P, R) ≤|F0 (α)2 f0 (α)2s |dα.
0The starting point in the iterative process is to bound the latter expressionin terms of Ss (Q1 , R) and T (1, s). This corresponds to taking the firstdifference in the classical Weyl differencing argument, and extracting thecontribution arising from those terms with x1 = y1 . Thus one obtains
Ss+1 (P, R)  P ε M12s−1 (P M1 Ss (Q1 , R) + T (1, s)),Waring’s Problem: A Survey

15

and this inequality we write symbolically as
F02 f02s 7−→ F1 f12s .
One way to proceed is by means of a repeated efficient differencing step.
In principle this is based on the Cauchy-Schwarz inequality, applied in the
form
Z 1
1/2 Z 1
1/2
Z 1
2s
2t
2 4s−2t|Fj fj |dα ≤|fj | dα
|Fj fj
|dα
,
0

0

0

where for the sake of concision we have written fj for fj (α) and likewise Fj
for Fj (α). Thus, for j = 1, 2, . . ., the mean value T (j, s) can be related to
St (Qj , R) and T (j + 1, 2s − t), where t < 2s is a parameter at our disposal,
via inequalities of the shape
ej M
fj M 4s−2t−1 Ξj+1 )1/2 ,T (j, s)  P ε (St (Qj , R))1/2 (Hj+1

where we write

ej M
fj+1 S2s−t (Qj+1 , R) + T (j + 1, 2s − t).
Ξj+1 = P HThis is the (j + 1)-th step in the differencing process and can be portrayedby
Fj fj2s4s−2t
Fj+1 fj+1


y

−→

fj2t

There are more sophisticated variants of this procedure wherein it may
be useful to restrict some of the variables to a range ( 21 Qj R−j , Qj ], or to
replace the set A(Qj , R) by N∩[1, Qj ] (see §2 of Vaughan and Wooley [134]for details, and a more complete discussion).
Another option is to use Hölder’s inequality to bound T (j, s). Thus we
obtain an inequality of the type
Z 1b
T (j, s) =Fj (α)fj (α)2s dα  Ila Il+1
Uvc Uwd ,
0

where
Im =

Z 1

2m

|Fj (α)|dα(m = l, l + 1),

0

Uu =

Z 1
0

2u

|fj (α)|dα(u = v, w),16

R. C. Vaughan and T. D. Wooley

and l, v, w, a, b, c, d are non-negative numbers satisfying the equations
a + b + c + d = 1,2l a + 2l+1 b = 1,

vc + wd = s.

There is clearly great flexibility in the possible choices of the parameters
here. We can summarise this process byl

Fj fj2s =⇒ (Fj2 )a (Fj2

l+1

)b (fj2v )c (fj2w )d .

Yet another option is to apply the Hardy-Littlewood method to T (j, s).
In practice we expect that the minor arc contribution dominates, although
this is not guaranteed. But if it does, then
T (j, s)  sup |Fj (α)| Ss (Qj , R),(12.1)
α∈m

and this we abbreviate to
Fj fj2s =⇒ (Fj )(fj2s ).
By optimising choices for the parameters in order to obtain the sharpest
estimates at each stage of the iteration process, one ultimately obtains relations describing λ(n+1) in terms of λ(n) . The sharpest permissible exponents λ attainable by these methods are in general not easy to describe, andrequire substantial computations to establish (see, for example, Vaughan
and Wooley [134]). However, one can describe in general terms the salientfeatures of the permissible exponents λs . When s is rather small comparedto k, it transpires that permissible exponents λs = s+δs can be derived withδs positive but small (see the next section for a consequence of this fact).
Further, the simplest versions of the repeated efficient differencing method(see Wooley [155][157]) establish that the exponent λs = 2s − k + ke1−2s/k
is permissible for every natural number s. Roughly speaking, therefore, one
may compare the respective strengths of the diminishing ranges argument,
and the repeated efficient differencing method, by comparing how rapidly
the respective functions k(1 − 1/k)s ∼ ke−s/k and ke1−2s/k tend to zero ass increases.
The improvements in the most recent work (Vaughan and Wooley [134])
come about mostly through the following technical improvements:• Better use of the Hardy-Littlewood method to estimate
Z 1
T (j, s) =Fj (α)fj (α)2s dα.
0

In particular, tighter control is exercised in mean on the behaviour of the
exponential sum Fj (α) on the major arcs, and this permits the assumptionof (12.1) for a larger range of parameters φ than previously available.

Waring’s Problem: A Survey

17• Better estimates for
Im =

Z 1

2m

|Fj (α)|dα,0established largely via estimates for the number of integral points on certain
affine plane curves.
• Better estimates (see Wooley [159]) for
sup |f0 (α)|.

α∈mSuch estimates might be described as playing a significant rôle in the estimation of G(k) in the final stages of the analysis.

13

Breaking Classical Convexity

All of the methods thus far described depend, in a fundamental manner, on
the natural interpretation of even moments of exponential sums in termsof the number of solutions of certain underlying diophantine equations. In§12, for example, one is limited to permissible exponents λs corresponding
to integral values of s, and in this setting the most effective method forbounding odd and fractional moments of smooth Weyl sums is to applyHölder’s inequality to interpolate between even moments. With the naturalextension of the notion of a permissible exponent λs from integral values ofs to arbitrary positive numbers s, the resulting exponents form the convex
hull of the set of permissible exponents {λs : s ∈ N}. A perusal of §12
reveals that extra flexibility in choice of parameters, and therefore thepotential for further improvements, will be achieved by the removal of this“classical convexity” barrier, and such has recently become available.
In Wooley [158], a method is established which, loosely speaking, enables one to replace the inequality
Ss+1 (P, R)  P ε M12s−1 (P M1 Ss (Q1 , R) + T (1, s))that occurred in §12 with s restricted to be a natural number, by the new
inequality
Ss+t (P, R)  P ε M12s−t (P t M1t Ss (Q1 , R) + T ∗ ),
where
T∗ =

Z 1
0

|F (α)t f1 (α)2s |dα18

R. C. Vaughan and T. D. WooleyandF (α) =X

Xe(αu−k (z1k − z2k )).

u∈A(M1 R,R) z1 ,z2 ∈A(P,R)
u>M1z1 ≡z2 (mod uk )z1 6=z2

In this latter estimate, the parameter s is no longer restricted to be integral,and the parameter t may be chosen freely with 0 < t ≤ 1. Moreover, themean value T ∗ is very much reminiscent of T (1, s), with F (α) substituted
for F1 (α) and exhibiting similar properties. Thus, in addition to removing the integrality constraint on s, one may also iterate with a fractional
number 2t of variables.
As might be expected, the additional flexibility gained in this way leads
to improved permissible exponents λs even for integral s, since our methods
are so highly iterative. The overall improvements are usually quite small
and are largest for smaller values of k. Such progress has not yet delivered
sharper bounds for G(k), but this work provides the sharpest results available concerning sums of cubes (see §4, and also Brüdern and Wooley [18]),and has also permitted new conclusions to be derived in certain problems
involving sums of mixed powers (see Brüdern and Wooley [16] [19] [17],and also §15 below). Also, this “breaking convexity” device provides thebest available lower bounds for the number Nk,s (X) of natural numbers
not exceeding X that are the sum of s kth powers of positive √
integers, atleast when s is small compared to k. Thus, when 2 < s ≤ 2e−1 k, one has
Nk,s (X)  X s/k−e

−γs k

,

where γs = 16/(es)2 (see Theorems 1.3 and 1.4 of Wooley [158], and theassociated discussion). For comparison, one conjecturally has the lower
bound Nk,s (X)  X s/k whenever s < k.
Much remains to be investigated for fractional moments, in part owing
to the substantial increase in complexity of the underlying computations(see Wooley [161] for more on this). However, such developments presently
appear unlikely to have a large impact on the central problem of bounding
G(k) in the classical version of Waring’s problem.

14

Variants of Waring’s Problem: Primes

Much work has been devoted to various generalisations of the classicalversion of Waring’s problem, and it seems appropriate to discuss some of
the more mainstream variants.
We begin with the Waring-Goldbach problem, in which one seeks torepresent integers as sums of kth powers of prime numbers. In order toWaring’s Problem: A Survey

19

describe the associated local conditions, suppose that k is a natural number
and p is a prime number. We denote by θ = θ(k, p) the integer with pθ |kand pθ+1 - k, and then define γ = γ(k, p) by(
θ + 2, when p = 2 and θ > 0,γ(k, p) =θ + 1, otherwise.
Finally, we put
K(k) =

Ypγ .

(p−1)|kDenote by H(k) the least integer s such that every sufficiently large positive
integer congruent to s modulo K(k) may be written as a sum of s kth
powers of prime numbers. Note that when (p − 1)|k, one has pθ (p − 1)|k,whence ak ≡ 1 (mod pγ ) whenever (p, a) = 1. This explains our seemingly
awkward definition of H(k), since whenever n is the sum of s kth powers
of primes exceeding k + 1, then necessarily n ≡ s (mod K(k)). Naturally,further congruence conditions may arise from primes p with (p − 1) - k.
Following the pioneering investigations of Vinogradov [145][146] (seealso Vinogradov [147]), Hua comprehensively investigated additive problems involving prime numbers in his influential book (see Hua [79], butalso Hua [78]). Thus, it is known that for every natural number k one has
H(k) ≤ 2k + 1,
and, when k is large, that
H(k) ≤ 4k(log k + 12 log log k + O(1)).
In the conventional plan of attack on the Waring-Goldbach problem,
one applies the Hardy-Littlewood method in a manner similar to that outlined above, but in interpreting the number of solutions of an analogueof the equation (11.1) over prime numbers, one obtains an upper boundby discarding the primality condition. With sufficiently many variables
employed to save a factor of n via such an approach, one additional variable suffices to save the extra power of log n required by primality considerations. Although this strategy evidently prohibits the use of smooth
numbers, the diminishing ranges technology perfected by Davenport, andrefined by Vaughan [121] and Thanigasalam [109]–[113] plays a prominentrôle in establishing the best available upper bounds for H(k) when k issmall. We should also mention that recent progress depends on good estimates of Weyl-type for exponential sums over primes, and allied sums,
available from the use of Vaughan’s identity (see Vaughan [117]), combined

20

R. C. Vaughan and T. D. Wooley

with the linear sieve equipped with a switching principle (see Kawada and
Wooley [83]). Thus, for 4 ≤ k ≤ 10, the best known upper bounds for H(k)are as follows:
Kawada and Wooley [83], H(4) ≤ 14, H(5) ≤ 21,
Thanigasalam [111], H(6) ≤ 33, H(7) ≤ 47, H(8) ≤ 63,
H(9) ≤ 83, H(10) ≤ 107.
Despite much effort on the Waring-Goldbach problem for exponents 1,
2 and 3, further progress remains elusive. Improvements are feasible, however, if one is prepared to accept almost-primes in place of prime numbers(see, in particular, Chen [25], Brüdern [13][14], and Brüdern and Fouvry[15]). Difficulties related to those associated with the Waring-Goldbach
problem are encountered when other sequences are substituted for prime
numbers. For Waring’s problem with smooth variables, see Harcos [57]and Brüdern and Wooley [18]. Also, see Nechaev [97] for work on Waring’s
problem with polynomial summands (Wooley [160] and Ford [55] have restricted improvements employing smooth numbers).

15

Variants of Waring’s Problem: Sums of Mixed
PowersSuppose that k1 , k2 , . . . , ks are natural numbers with 2 ≤ k1 ≤ k2 ≤ · · · ≤kt . Then an optimistic counting argument suggests that whenever theequation
xk11 + xk22 + · · · + xks s = n(15.1)
has p-adic solutions for each prime p, andk1−1 + k2−1 + · · · + ks−1 > 1,

(15.2)

then n should be represented as the sum of mixed powers of positive integers(15.1) whenever it is sufficiently large in terms of k. When s = 3 such an
assertion may fail in certain circumstances (see Jagy and Kaplansky [80],or Exercise 5 of Chapter 8 of Vaughan [128]), but a heuristic application of
the Hardy-Littlewood method suggests, at least, that the condition (15.2)
should ensure that almost all integers in the expected congruence classes
are thus represented. Moreover, subject instead to the condition
k1−1 + k2−1 + · · · + ks−1 > 2,

(15.3)

a formal application of the circle method suggests that all integers in the expected congruence classes should be represented in the form (15.1). Meanwhile, a simple counting argument shows that in circumstances in which

Waring’s Problem: A Survey

21

the condition (15.2) does not hold, then arbitrarily large integers are notrepresented in the desired form.
The investigation of such analogues of Waring’s problem for mixed powers has, since the early days of the Hardy-Littlewood method, stimulated
progress in technology of use even in the classical version of Waring’s problem. Additive problems in which the summands are restricted to be squares,
cubes or biquadrates are perhaps of greater interest than those with higher
powers, and here the current situation is remarkably satisfactory. We summarise below the current state of knowledge in the simpler problems of thisnature. In Tables 5 and 6 we list constellations of powers whose sum represents, respectively, almost all, and all, integers subject to the expectedcongruence conditions. The tables are arranged, roughly speaking, starting with predominantly smaller exponents, and ending with predominantly
larger exponents, and therefore not in chronological order of the results.
We have been unable to trace the origin in the literature of the conclusion on a square, two biquadrates and a kth power, but refer the reader
to Exercise 6 of §2.8 of Vaughan [128] for related ideas (see also RothDavenport & Heilbronn [36],Davenport & Heilbronn [37],Roth [101],
Vaughan [116],
Folklore (?)
Hooley [70],
Davenport [29],Brüdern [9][8],Brüdern [8], Lu [89],Brüdern & Wooley [19],Kawada & Wooley [82],Vaughan [124],
Kawada & Wooley [82],two squares, one kth power,
one square, two cubes,
one square, one cube, one biquadrate,
one square, one cube, one fifth power,
one square, two biquadrates,
one kth power,
one square, one cube, one sixth power,
one kth power,
four cubes,
three cubes, one biquadrate,
three cubes, one fifth power,
three cubes, one sixth power ,one cube, four biquadrates,
six biquadrates,
five biquadrates, one kth power(k odd).

Table 5. Representation of almost all integers

22

R. C. Vaughan and T. D. Wooley

Gauss [56],
Hooley [71],
Hooley [70],Vaughan [123],Brüdern & Wooley [16],Ford [54],Linnik [87][88],Brüdern [9],Brüdern [9],Kawada & Wooley [82],Brüdern & Wooley [17],Kawada & Wooley [82],Vaughan [124],
Kawada & Wooley [82],three squares,
two squares, three cubes,
two squares, assorted powers,
one square, five cubes,
one square, four cubes, one biquadrate,
one square, one cube, one biquadrate,...,
one fifteenth power,
seven cubes,
six cubes, two biquadrates,
five cubes, three biquadrates,
three cubes, six biquadrates,
two cubes, seven biquadrates,
one cube, nine biquadrates
twelve biquadrates,
ten biquadrates, one kth power(k odd).

Table 6. Representation of all integers[101]). We remark that all ternary problems of interest have been solved,
since for non-trivial triples (k1 , k2 , k3 ) not accounted for in Table 5, one has
k1−1 +k2−1 +k3−1 ≤ 1. Also, energetic readers may be interested in tackling aproblem which presently defies resolution only by the narrowest of margins,
namely the problem of showing that almost all integers are represented as
the sum of two cubes and two biquadrates of positive integers.
Note that although the three square theorem is commonly ascribed toLegendre, his “proof” depended on an unsubstantiated assumption only
later established by Dirichlet, and the first complete proof is due to Gauss.
We finish by noting that in problems involving sums of two squares, methods more effective than the circle method can be brought into play (seeespecially Hooley [70][71] and Brüdern [7]).

16

Variants of Waring’s Problem: Beyond Z

Given the considerable energy expended on the investigation of Waring’s
problem over the rational integers, it seems natural to extend this workWaring’s Problem: A Survey

23

to algebraic number fields. Here one encounters the immediate difficultyof deciding what precisely Waring’s problem should mean in this broader
context. It is possible, for example, that an algebraic integer in a number
field K may not be a sum of any finite number of kth powers of algebraic
integers of that field (consider, say, the parity of the imaginary part of a2
when a ∈ Z[i]). With this in mind, when K is a number field and OK
is its ring of integers, we define Jk to be the subring of OK generated by
the kth powers of integers of K. We must also provide an analogue of thepositivity of the kth powers inherent in the classical version of Waring’s
problem. Thus we define GK (k) to be the smallest positive integer s withthe property that, for some positive number c = c(k, K), and for all totally
positive integers ν ∈ Jk of sufficiently large norm, the equation
ν = λk1 + λk2 + · · · + λks(16.1)is always soluble in totally non-negative integers λj of K with N (λj ) ≤cN (ν)1/k (1 ≤ j ≤ s).
Following early work of Meissner [93] and Mordell [95] for a restrictedclass of number fields, Siegel [102][103] was the first to obtain quite general
conclusions for sums of squares, and hence, via the method of Hilbert [69],for sums of kth powers. Siegel later developed a proper generalisation of
the Hardy-Littlewood method to number fields, and here the dissection
into major and minor arcs is a particular source of difficulty. In this way,
Siegel [104][105] obtained the upper boundGK (k) ≤ dk(2k−1 + d) + 1,
where d denotes the degree of the field K. If one were to break the equation (16.1) into components with respect to an integral basis for OK , then
one would obtain d equations of degree k in ds variables, and so one might
optimistically expect the analytic part of the circle method to apply with anumber of variables roughly the same in both K and Q. Perhaps motivatedby such considerations, Siegel asked for reasonable bounds on GK (k) independent of d. This question was ultimately addressed through the work of
Birch [4] and Ramanujam [98], who provided the upper boundGK (k) ≤ max{8k 5 , 2k + 1}.

(16.2)

It is evident that the uniform bound (16.2) is far from the above citedbound G(k) ≤ (1 + o(1))k log k of Wooley [155], and the only slightly
weaker precursors of Vinogradov. An important desideratum, therefore, isthe reduction of the bound (16.2) to one of similar order to that presently
available for G(k), or at least a reduction to a bound polynomial in k.

24

R. C. Vaughan and T. D. Wooley

Failing this, effort has been expended in pursuit of bounds of order k log k,
but with a modest dependence on the degree d of the field K. Progress
towards this objective has mirrored developments in the classical version of
Waring’s problem. Thus, building on work generalising that of Vinogradovto the number field setting by Körner [84] and Eda [50], methods employing smooth numbers and repeated efficient differences have recently been
applied by Davidson [39] to establish the boundGK (k) ≤ (3 + o(1))k log k + cd k,
where
cd = 4d + 3 log( 21 (d2 + 1)) + 7,
and the term o(1) is independent of d (apparently, the term 3 + o(1) here
can be replaced by 2 + o(1) with only modest effort). Moreover, when K
has class number one, Davidson [40][42] has obtained the boundGK (k) ≤ k(log k + log log k + cd ),where cd is approximately 4d. Finally, Davidson [41] has improved onearlier work of Tatuzawa [106] to establish the strikingly simple conclusion
that, for every number field K, one hasb
GK (k) ≤ 2d(G(k)+ 2k),bwhere G(k)denotes the least number s satisfying the property that theset of rational integers that can be expressed as the sum of s kth powersbof natural numbers has positive density (in particular, of course, G(k)≤G1 (k)).
More exotic still than the variants of Waring’s problem over number
fields are those in which one works over the polynomial rings Fq [t]. Hereone must again impose restrictions on the size of polynomials employed in
the representation (and in this situation, the degree of a polynomial provides a measure of its size). Analogues of the Hardy-Littlewood method
have been devised in this polynomial ring setting (see, for example, Effinger
and Hayes [51]). Unfortunately, however, Weyl differencing proves ineffective whenever kth powers are considered over Fq [t] with k ≥ char(Fq ), forin such circumstances the factor k!, introduced into the argument of the
exponential sum over kth powers via the differencing argument, is equal tozero in Fq [t]. Consequently, one frequently restricts attention to kth powers with k smaller than the characteristic (but see Car and Cherly [22] forresults on sums of 11 cubes in F2h [t]). With this restriction, Car [20][21]has shown that every polynomial M , with M ∈ Fq [t], of sufficiently large
degree, can be written in the form
M = M1k + · · · + Msk ,Waring’s Problem: A Survey

25

with Mi ∈ Fq [t] of degree smaller than 1 + (deg(M ))/k (1 ≤ s ≤ k),provided thats ≥ min{2k + 1, 2k(k − 1) log 2 + 2k + 3}.

17Open Problems and Conjectures

Returning temporarily to the methods of §12, there are a number of problems connected with the mean value
Z 1
m
Im =
|Fj (α)|2 dα0which suggest some interesting questions. We will concentrate on the situation with m = 1, and remark only that the available results become lesssatisfactory as m increases.
A simple combinatorial argument reveals that the difference polynomialΨj = Ψj (z; h; m) is given explicitly by the formulaΨj = k!2j h1 . . . hj

XX

···

u≥0 v1 ≥0

X z u (h1 mk1 )2v1 . . . (hj mkj )2vj

vj ≥0

u!(2v1 + 1)! . . . (2vj + 1)!

,

where the summation is subject to the condition u + 2v1 + · · · + 2vj = k − j.
Consequently, one has
12 (k−j−d)Ψj = h1 . . . hj zdX

cr (h1 mk1 , . . . , hj mkj )z 2r ,

r=0

where

(
0, when k − j is even,d=
1, when k − j is odd,

and for 0 ≤ r ≤ (k − j − d)/2, the coefficients cr (ξ) are polynomials withpositive integral coefficients that are symmetric in ξ12 , . . . , ξj2 , and have total
degree k − j − 2r − d. Now the mean value
I1 =

Z 1

|Fj (α)|2 dα0

is equal to the number of solutions of the diophantine equation
Ψj (z; h; m) = Ψj (z 0 ; h0 ; m0 ),(17.1)

26

R. C. Vaughan and T. D. Wooley

with the variables in ranges discernible from the definition of Fj (α) in§12, and one might hope that the total number of solutions is close to thenumber of diagonal solutions, which is to say that
fj H
ej .
I1  P 1+ε M

When k−j is odd (so d = 1), the presence of the term z d makes it especiallyeasy to deal with equation (17.1) by exploiting the inherent multiplicative
structure, and indeed one can achieve the desired bound provided also thatj ≤ (k − d)/3. The cases j = 1 and k − j = 2 or 4 are also doable.
However, when k − j is even and 1 < j ≤ k − 6, the situation is not soeasy. By the way, this difficulty already occurs in Davenport’s work [31].
To illustrate this situation, the simplest special case that we cannot handle
directly corresponds to k = 8 and j = 2, and here one has
2 16Ψj = h1 h2 224z 6 + 1120z 4 (h21 m16
1 + h2 m2 )2 2 16 16
4 32+ z 2 (672h41 m32
1 + 2240h1 h2 m1 m2 + 672h2 m2 )


4 2 32 16
2 4 16 32
6 48+ 32h61 m48
1 + 224h1 h2 m1 m2 + 224h1 h2 m1 m2 + 32h2 m2 .
This suggests various general questions.
• Suppose that f, g ∈ Z[x]. Are there simple conditions on f, g such thatthe number N of integral points (x, y) ∈ [−P, P ]2 for which f (x) = g(y)satisfiesεN  P H(f )H(g) ?
Here H(h) denotes the height of h. A qualitative version of this has alreadybeen considered by Davenport, Lewis and Schinzel [38], and if f (x) − g(y)is irreducible over C, then a celebrated theorem of Siegel shows that the
number of solutions is finite unless there is a rational parametric solutionof special form.
By the way, in view of the above, it is perhaps not surprising that in
our treatment of Im the bound of Bombieri and Pila [6] plays a rôle.
• Suppose that A ⊂ Zk ∩ [−X, X]k . Let R(n; A) denote the number ofsolutions of the equation
a1 x + a2 x2 + · · · + ak xk = n
with x ∈ Z ∩ [−P, P ] and a ∈ A. Are there any simple conditions underwhich it is true that
X
R(n; A)2  card(A)P (XP )ε ?
n

Waring’s Problem: A Survey

27• A well-known conjecture in connection with Waring’s problem is Hypothesis K (Hardy and Littlewood [62]). Let
Rk,s (n) = card{x ∈ Ns : xk1 + · · · + xks = n}.
Then Hypothesis K asserts that for each natural number k, one has
Rk,k (n)  nε .

(17.2)

From this it would follow that
G(k) ≤ max{2k + 1, Γ0 (k)}(17.3)

G1 (k) = max{k + 1, Γ0 (k)}.

(17.4)

andThe conjecture (17.2) was later shown by Mahler [90] to be false for k =3, and indeed his counter-example shows that, infinitely often, one has
R3,3 (n) > 9−1/3 n1/12 . However, the conjecture is still open when k ≥ 4,
and for (17.3) and (17.4), it suffices to know that
X
Rk,k (n)2  N 1+ε .
n≤N

Hooley [74] established this when k = 3, under the assumption of the
Riemann Hypothesis for a certain Hasse-Weil L-function. As far as weknow, no simple conjecture of this kind is known from which it would
follow that G(k) = max{k + 1, Γ0 (k)}.
• It may well be true that, when k ≥ 3, one has
X
Rk,k (n)2 ∼ CN.
n≤N

However Hooley [72] has shown, at least when k = 3, that the constant C
here is larger than what would arise simply from the major arcs. This leads
to some interesting speculations. The number of solutions of the equation
xk1 + · · · + xks = y1k + · · · + ysk ≤ N,
in which the variables on the right hand side are a permutation of those on
the left hand side, is asymptotic to C1 N s/k , for a certain positive number
C1 = C1 (k, s), and when s < k the contribution arising from the majorarcs is smaller. Maybe one should think of these solutions as being “trivial”, “parametric”, or as arising from some “degenerate” property of thegeometry of the surface. Anyway, their contribution is mostly concentrated

28

R. C. Vaughan and T. D. Wooleyon the minor arcs. It seems rather likely that this phenomenon persists fors ≥ k and explains Hooley’s discovery. This leads to the philosophy thatthe major arcs correspond to non-trivial solutions, and the minor arcs totrivial solutions. There is an example of this phenomenon in Vaughan and
Wooley [133].
• Recall the definitions of f (α), S(q, a) and v(β) from §7. One can conjecture that, whenever (a, q) = 1, one has
1/kf (α) − q −1 S(q, a)v(α − a/q)  q + P k |qα − a|.

(17.5)

Possibly the exponent has to be weakened to 1/k + ε, but any counterexamples would be interesting.
From (17.5) it would follow that
X
Rk,k (n)2  N.
n≤NAlso it is just conceivable that (17.5), in combination with a variant of the
Hardy-Littlewood-Kloosterman method, would achieve the bound G(k) ≤max{G(k), Γ0 (k)}, where G(k) < 2k + 1.
• The inequality (17.5) is a special case of conjectures that can be madeabout the exponential sumXf (α) =e(α1 x + · · · + αk xk )x≤Pthat would have many consequences in analytic number theory. For example one can ask if something likef (α) − q −1 S(q, a)v(α − a/q) 

1/j
k 
X
q + P j |αj q − aj |j=2

is true. Here(q, aj )qXe((a1 r + · · · + ak rk )/q),Z Pe(β1 γ + · · · + βk γ k )dγ,

S(q, a) =(17.6)

r=1

v(β) =

0and a1 = c(q, a2 , . . . , ak ), where c = c(q, a) is the unique integer with−1
1< c − α1 q/(q, a2 , . . . , ak ) ≤ .
22

Waring’s Problem: A Survey

29

The inequality (17.6) is known to hold for k = 2 (Vaughan unpublished)with the right hand side weakened slightly toq(q, a2 )1/2 


2q
1/2
log
+ P |α2 − a2 /q2 |.
(q, a2 )• One way of viewing the Hardy-Littlewood method is that we begin byconsidering the Fourier transform with respect to Lebesgue measure on[0, 1) for an appropriate generating function defined in terms of the additive
characters, and then approximate to it by a product of discrete measures.
Since part of the problem when one has relatively few variables is that thegeometry genuinely intrudes, one can ask whether we are using the best
measure for the problem at hand. In this situation something more closely
related to the underlying geometry might be more useful.
In conclusion, it is clear that although we have come a long way in thetwentieth century, there remains plenty still to be done!

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