# Near-optimal mean value estimates

for multidimensional Weyl sums

###### Abstract.

We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.

###### Key words and phrases:

Exponential sums, Hardy-Littlewood method, Diophantine equations###### 2010 Mathematics Subject Classification:

11L15, 11L07, 11D45, 11D72, 11P55## 1. Introduction

The investigation of Diophantine problems of large degree is in general fraught with difficulties only partially mollified by the presence of intrinsic diagonal structure. Indeed, such analyses as are made available via the Hardy-Littlewood (circle) method, when successful, involve complicated exponential sum estimates widely considered to be amongst the most challenging in the subject. The application of Weyl differencing by Davenport, Birch and Schmidt, on the one hand, involves a delicate interplay between the singular locus associated with the problem and the quality of ensuing exponential sum estimates (see [4], [5], [13]). As an inescapable feature of such approaches, the number of variables required in a successful treatment grows exponentially with the degree of the problem at hand. The extension of Vinogradov’s methods to exponential sums in many variables, on the other hand, is notoriously complicated. The work of Arkhipov, Karatsuba and Chubarikov [1], [2], for example, permits substantially sharper conclusions to be drawn when partial diagonal structure is present. However, the complexity of the underlying methods has deterred a consideration of all but the simplest model situations (see [2] and [9]). In addition, the available conclusions fail to achieve their conjectured potential by a factor growing roughly like the logarithm of the total degree of the associated translation-invariant Diophantine system.

Our goal in this paper is to extend the efficient congruencing method introduced by the third author [21] so as to accommodate the generalised Vinogradov systems of Arkhipov, Karatsuba and Chubarikov (see [1], [2]). It transpires that for systems of large degree, the bounds that we thereby derive miss those conjectured to hold by a factor of only or thereabouts, transforming the previous state of the art. Moreover, our methods are of such flexibility that they may be successfully applied to translation-invariant systems of wide generality, and in particular to systems closely related to those subject to recent investigations by quantitative arithmetic geometers studying the Manin-Peyre conjectures (see [14, §4.15], [15]). Since our methods yield estimates no less striking for such systems, we take the opportunity to derive rather general estimates again coming within a constant factor of those conjectured to hold. There are consequences of all of this work for exponential sum estimates of Weyl-type, for the solubility of systems of Diophantine equations and related problems, and for certain problems in additive combinatorics, and these we also explore herein.

Rather than encumber the reader at this point with the substantial notational prerequisites entailed by a discussion of our most general conclusions, we instead offer the more easily digestible corollaries particular to the model problem considered in earlier work [9] of the first author. Let , and be natural numbers, and let be a positive number. We focus attention on the system of simultaneous Diophantine equations

(1.1) |

Here, the indices are non-negative integers, so that a modest computation reveals the total number of equations in the system (1.1) to be , where

(1.2) |

Meanwhile, the total degree of the system (1.1), which is to say the sum of the degrees of the equations comprising the system, is equal to , where

(1.3) |

In particular, in the familiar classical Vinogradov system with , one has and . Finally, we write for the number of integral solutions of the system (1.1) with .

In §2, as a special case of Theorem 2.1, we derive an estimate for that is in many respects close to the best possible. Here and throughout, implicit constants in Vinogradov’s notation and depend at most on , , and , unless otherwise indicated. The letter should be interpreted as a positive number sufficiently large in terms of , , and .

###### Theorem 1.1.

Suppose that , and are natural numbers with and . Then for each , one has .

The special case of Theorem 1.1 with is equivalent to [21, Theorem 1.1], a conclusion which has very recently been sharpened in [22, Theorem 1.1], so that for and , one has

When , meanwhile, one may compare the conclusion of Theorem 1.1 with work of Arkhipov, Karatsuba and Chubarikov [2]. Although set up slightly differently, it is clear that the methods of the latter authors have the potential to establish a bound of the shape

where decays with roughly like . In [9, Theorem 1.1] this conclusion was improved by the first author for sufficiently large values of , so that on writing

one may take

The estimate supplied by Theorem 1.1 is substantially sharper. Thus, provided only that , one may take for any positive number . The number of variables required in typical applications, as we discuss in due course, is thereby reduced by a factor of order .

In order to discern the strength of the estimate supplied by Theorem 1.1, we must consider available lower bounds for the mean value , and thereby infer plausible conjectures for corresponding upper bounds. In §3 we establish the lower bound for contained in the following theorem.

###### Theorem 1.2.

Suppose that , and are natural numbers. Then one has

It seems reasonable to conjecture that whenever , one has the allied upper bound

Thus, when is sufficiently large in terms of and , one expects that

(1.4) |

as is confirmed by Theorem 1.1 for . We emphasise that here, and throughout the introduction, we abbreviate to and to . As a consequence of Theorem 1.2, we show in Theorem 3.2 that when is a real number with , and

then there is a positive number with the property that

When , therefore, it follows that the conclusion of Theorem 1.1 comes within a factor of the least value of for which the conjectured upper bound (1.4) might conceivably hold. In such multidimensional Weyl sums, a near-optimal conclusion of this type, merely a constant factor away from the best possible, has hitherto been wholly beyond our grasp.

We next consider upper bounds for exponential sums of Weyl-type, the discussion of which is much facilitated by the introduction of additional notation. It is convenient to abbreviate a monomial of the shape to , in which . Likewise, we may write for . In such circumstances, we put

Also, in place of the -tuple we write , and we adopt the convention that is to mean that each coordinate of the vector satisfies . Equipped with these conventions, the Diophantine system (1.1) assumes the compact shape

and counts the number of integral solutions of this system with .

Define the exponential sum by

where

and, as usual, we write for . Here, the subscript that identifies as the -tuple may usually be omitted without leading to confusion. As we have already noted, the number of coefficients is . We adopt the convention that, when is measurable, then

It then follows from orthogonality that

(1.5) |

Upper bounds for mean values of exponential sums such as may be converted into Weyl-type estimates by means of variants of the large sieve inequality. Before announcing such an estimate, which is a consequence of the more general result recorded in Theorem 10.3, we pause to record a further notational convention. When , we write for the greatest common divisor .

###### Theorem 1.3.

Suppose that and are natural numbers with . Let be any real number with

Then whenever , for some , it follows that there exist and satisfying

The special case of Theorem 1.3 with is very slightly weaker than [21, Theorem 1.6], a conclusion which has recently been sharpened in [22, Theorem 11.2]. Thus, when , the conclusion of Theorem 1.3 holds whenever and . The work of Arkhipov, Karatsuba and Chubarikov [2], as interpreted and sharpened by the first author, yields a conclusion similar to Theorem 1.3. Indeed, it follows from a corrected version^{1}^{1}1See the discussion following the proof of Theorem 10.2 below for an explanation of the need for a modest correction in [9, Theorem 1.2]. of [9, Theorem 1.2] that a conclusion of similar form holds for sufficiently large values of , though with the constraint on the exponent replaced by a condition of the shape . On noting that , the superiority of our new bound is clear.

We next consider the application of our new estimates to Diophantine problems. When , and are natural numbers, and is a non-zero integer for and , write

In §11, we consider the Diophantine system

(1.6) |

consisting of equations of total degree . Let denote the number of integral solutions of the system (1.6) with . We follow Schmidt [13] when defining the (formal) real and -adic densities associated with the system (1.6). When , define

and put

The limit , when it exists, is called the real density. Meanwhile, given a natural number , we write

For each prime number , we then put

provided that this limit exists, and refer to as the -adic density.

As a special case of Theorem 11.1, we establish an asymptotic formula for valid whenever .

###### Theorem 1.4.

We note that [21, Theorem 9.1] delivers the same conclusion as Theorem 1.4 in the special case . As is apparent from the lower bound supplied by Theorem 1.2 and the ensuing discussion, there exist choices of coefficients for which the asymptotic formula (1.7) necessarily fails when is large and

(1.8) |

Consequently, the bound on the number of variables in the hypotheses of Theorem 1.4 is within a factor of the best possible bound for such systems. Indeed, the argument underlying the proof of Theorem 1.2 shows that such remains true in wider generality. The point here is that special subvarieties contain the bulk of the set of integral solutions whenever the bound (1.8) holds on the number of blocks of variables. Conclusions available hitherto of the type presented in Theorem 1.4 impose bounds on the number of blocks of variables weaker than our own by a factor of order .

As a special case of Corollary 11.2, we obtain an asymptotic formula for the mean value .

###### Theorem 1.5.

Let and be natural numbers with . Then whenever , there exist positive constants and such that

A conclusion analogous to that of Theorem 1.5 is obtained in [9, Theorem 1.3], subject to the condition that be sufficiently large and

Again, the conclusion of Theorem 1.5 is much superior. When and , meanwhile, the conclusion of Theorem 1.5 is a consequence of [21, Theorem 1.2].

As a penultimate application of the bounds supplied by Theorem 1.1, in §11 we consider the rational linear spaces of projective dimension lying on the diagonal hypersurface

(1.9) |

with fixed non-zero integers. Such a linear space may be written in the form

for suitable linearly independent vectors . As noted in [9], by substituting into (1.9) and using the multinomial theorem to collect together coefficients of , one finds that the linear space corresponds to a solution of the Diophantine system

(1.10) |

This correspondence is made explicit by means of the simple relation

Write for the number of integral solutions of the system (1.10) with , and put

In §11 we indicate how to prove an asymptotic formula for subject to the condition that . In this context, we say that the integral -tuple is a non-singular choice of coefficients for and when the system of equations (1.10) has non-singular real and -adic solutions, for every prime number .

###### Theorem 1.6.

Suppose that , and are natural numbers with and . Suppose further that is a non-singular choice of coefficients for and . Then there exist positive constants and such that

In particular, one finds that whenever and appropriate local solubility conditions are met, then the hypersurface defined by (1.9) contains an abundance of rational linear spaces of projective dimension . A perusal of [9] reveals that, for sufficiently large values of , a similar conclusion is asserted by Theorem 1.4 of the latter source, subject instead to the more stringent condition

We remark that the lower bound in Theorem 1.6 should be susceptible to some small improvement by adapting the methods of [20] to the present multidimensional setting. Moreover, when the degree is very small, an approach of the first author [10] motivated by a method of Hua proves superior in some situations.

Further applications within the orbit of our methods and bounds include the generalised Waring problem of representing a given polynomial

in the form

and also results concerning the number of integral solutions of Diophantine inequalities modulo . We refer the reader to [2] for a discussion of some such problems, and leave to the reader the satisfaction of incorporating our new bounds into the established methods so as to make similarly striking improvements over the previous state of knowledge.

As a final application of our new bounds for multidimensional Weyl sums, we announce an application in additive combinatorics based on the second author’s recent work [11] on translation invariant systems of equations devoid of solutions in multidimensional sets. In Theorem 11.3 below we present a conclusion more general than the one we presently record in Theorem 1.7. For the purpose at hand, we describe the integral -tuple as an extended non-singular choice of coefficients for and when (i) one has , and (ii) the system of equations

(1.11) |

has non-singular real and -adic solutions, for every prime number .

Certain solutions of the system (1.11) are atypically simple to obtain, such as the trivial solutions lying on the diagonal . We formalise this notion by distinguishing two types of special solutions of (1.11). We describe as projected when there is a translate of a proper subspace of that contains all of . The aforementioned diagonal solutions are therefore projected, since they lie in a translate of the trivial subspace of . Also, we say that is a subset-sum solution when there exists a partition , into disjoint non-empty sets , such that for one has

In the special case in which

one sees that there are trivial subset-sum solutions in which whenever and .

###### Theorem 1.7.

Suppose that , and are natural numbers with and . Suppose further that is an extended non-singular choice of coefficients for and , so that . Let be a subset of , and suppose that the only solutions of the system (1.11) from are either projected or subset-sum solutions. Then one has

This theorem is a higher dimensional cousin of [11, Theorem 5.1], which supplies an analogous conclusion for a case involving binary forms. Theorem 1.7 shows that when grows more rapidly than , then the system (1.11) contains solutions from besides such obvious ones as the diagonal solutions with . As we see in §3, the extended system (1.11) contains more general special subvarieties defined by means of a projection process, the simplest of which set one or more variables to be zero. In Theorem 11.3 we present a conclusion that refines Theorem 1.7 in which, under the same hypotheses concerning the cardinality of , one finds that the system (1.11) contains solutions from which avoid all of these special subvarieties. In this way, one may legitimately describe the solutions of (1.11) thus shown to exist as honestly non-trivial. The interested reader will find the necessary ideas in earlier work [11] of the second author.

It may be useful to provide an informal sketch hinting at the argument underlying the proof of Theorem 1.1 so that the reader is better prepared to draw parallels with previous approaches. A more comprehensively illuminated sketch of this argument in the case may be found in [21, §2]. In common with the previous approaches of [2] and [9], the basic tool employed in our proof of Theorem 1.1 is a (so-called) -adic iteration mirroring the one devised by Linnik [6] in the classical setting with . Thus, we begin by artificially introducing a congruence condition, modulo a suitable prime , amongst the bulk of the variables underlying the mean value (1.5). An application of Hölder’s inequality leads to a new mean value in which the latter variables lie in common congruence classes across blocks. At this point, the multiple translation invariance of the system (1.1) may be utilised so as to pass to the zero congruence class, and thereby a congruence condition is forced on a subset of the variables of greater strength than that previously introduced. The approach of [2] is to choose the prime in such a way that this strong congruence condition forces a diagonal condition amongst blocks of variables, and thereby one is able to bound a mean value involving blocks of variables in terms of a corresponding mean value involving blocks of variables. In [9] the strong congruence condition is interpreted as a differencing process analogous to, though more efficient than, that of Weyl. By appropriate use of the Cauchy-Schwarz inequalities, one is able to repeat this efficient differencing process, deferring the moment at which to force the diagonal condition. In the present paper, following [21], we instead interpret the strong congruence condition as an efficient method of imposing a second artificial congruence condition amongst variables. By appropriate application of Hölder’s inequality, one recovers a new mean value resembling that obtained in the first step, but now yielding a fresh congruence condition amongst variables significantly stronger than before. If one begins with a mean value significantly larger in size than anticipated, then repeated application of this efficient congruencing procedure yields a related mean value larger in size than that anticipated by an amount so large that even a trivial estimate demonstrates the presumed initial deviation from the expected size to be untenable. In this way, one shows that the mean value under consideration has size very close to that expected.

We finish by emphasising that the methods of this paper are robust to changes of the ambient ring. Thus the rational integers central to this paper may be replaced with the ring of integers from a number field, or the polynomial ring , without diminishing the strength of the ensuing estimates. Such ideas have been explored very recently in the case in work emerging from the body of research exploiting the efficient congruencing method (see [7] and [23]).

In §2 we introduce the general translation-dilation invariant systems which constitute the central objects of attention in this paper. Then, in §3, we discuss the lower bounds recorded in Theorem 1.2. The notation and infrastructure required for our most general conclusions is discussed in §4, and then in §5 we derive the basic mean value estimates which initiate our efficient congruencing argument. Next, in §6, we provide estimates for the number of solutions of a system of basic congruences. Here, the singular locus of the system is of particular concern. The conditioning process, required to guarantee appropriate non-singularity conditions, is engineered in §7, and in §8 we discuss the efficient congruencing process itself. In §9 we combine the output of §§7 and 8 so as to deliver Theorem 1.1 via an iterative process. Consequences for Weyl-type estimates are discussed in §10, yielding the conclusion of Theorem 1.3. Finally, in §11 we sketch the arguments required to establish the Diophantine consequences recorded in Theorems 1.4-1.7.

## 2. Translation-dilation invariant systems

In order to describe our most general conclusions, we must introduce some notation having flexibility sufficient for our needs. An overly prescriptive approach has the potential to shroud the details of our arguments in a thick blanket of impenetrable symbols. With this undesirable potential outcome in mind, we opt for a somewhat abstract approach, and only later do we spend time detailing the most interesting situations.

Let , and be natural numbers, and consider a system of homogeneous polynomials , where . We investigate the system of Diophantine equations

(2.1) |

in which and for . Note that, in view of our conventions concerning vector notation, the system (2.1) consists of simultaneous Diophantine equations. Write and , and denote by the number of integral solutions of the system (2.1) with .

In this paper we are concerned with translation-dilation invariant systems of the shape (2.1). With the discussion to come in mind, we take a pragmatic approach to defining such systems. We say that the system is translation-dilation invariant if:(i) the polynomials are each homogeneous of positive degree, and(ii) there exist polynomials

with for , having the property that whenever , then

(2.2) |

Extend the definition of the coefficients by putting when . Then on writing and for the matrix , we see that the relations (2.2) are summarised by the formula

(2.3) |

Notice that the matrix is lower unitriangular, which is to say that it is a lower triangular matrix whose main diagonal entries are all . Suppose that is a natural number, that is a non-zero rational number, and . Then we see from (2.2) that the Diophantine system (2.1) possesses an integral solution if and only if one has

(2.4) |

This observation justifies the description of such systems of equations as trans-lation-dilation invariant. We should note that while this formal definition facilitates many of our arguments, it is clear that one may rearrange the ordering of the forms, and also consider independent linear combinations of the original forms, without altering the number of integral solutions of the system (2.1) counted by . Thus we may be expedient in most circumstances, and instead describe a system as translation-dilation invariant when it is equivalent in such a manner to some new system which is translation-dilation invariant in the strict sense.

We emphasise that translation-dilation invariant systems are easily generated. Given a collection of homogeneous polynomials

consider the set consisting of all the partial derivatives

(2.5) |

with . Plainly, when exceeds the largest total degree of any of the polynomials , this partial derivative vanishes. The set is consequently finite. Let denote the subset of consisting of all polynomials in having positive degree. We write , labelling the elements in such a way that . An application of the multidimensional version of Taylor’s theorem now shows that the relations (2.2) hold for some choice of coefficients satisfying . Since we may replace the set of forms by any subset whose span contains the polynomials , there is no loss of generality in supposing the set to be linearly independent. Such a system of forms we call reduced.

Finally, by replacing the forms by appropriate linear combinations of the original forms, we find that there is no loss of generality also in supposing that the matrix with entries is lower unitriangular. This new system , generated from the partial derivatives (2.5), is a reduced translation-dilation invariant system.

We are now almost equipped to state our main theorem, but first pause to introduce some parameters associated with a translation-dilation invariant system of polynomials . When consists of polynomials , we refer to the number of variables in as the dimension of the system. In addition, we describe the number of forms comprising as the rank of the system. We write for the total degree of the polynomial , and then define the degree of the system by

and the weight by

Our goal in §§4-9 is the proof of the following mean value estimate, which represents the main theorem of this paper.

###### Theorem 2.1.

Let be a reduced translation-dilation invariant system of polynomials having dimension , rank , degree and weight . Suppose that is a natural number with . Then for each , one has .

So far as we are aware, no mean value estimate available in the literature has generality to compete with Theorem 2.1. Moreover, when the estimates available hitherto are considerably weaker, even in the special situations in which they are applicable. In order to illustrate the ease with which estimates may be extracted from Theorem 2.1, we finish this section with a brief discussion of some simple cases, and in particular we show how to establish Theorem 1.1 as a consequence of Theorem 2.1.

(a) The classical system of Vinogradov [16], [17]. Consider the seed polynomial . By taking successive derivatives, we find that an associated reduced translation-dilation invariant system of polynomials is . This system has dimension , rank , degree and weight

Then it follows from Theorem 2.1 that when , one has

This estimate recovers the main conclusion of the third author’s recent work introducing the efficient congruencing method to Vinogradov’s mean value theorem (see [21, Theorem 1.1]). We note that subsequent work of the third author leads to the improved constraint on the number of variables in this conclusion (see [22, Theorem 1.1]).

(b) The system of Parsell [9]. Consider the situation with and seed polynomials . By taking successive partial derivatives, we find that an associated reduced translation-dilation invariant system of polynomials is

This system has dimension , rank

(2.6) |

degree and weight

(2.7) |

In this instance, it follows from Theorem 2.1 that when , one has . In view of (1.2) and (1.3), this completes the proof of Theorem 1.1.

(c) The system of Arkhipov, Karatsuba and Chubarikov [2]. Consider the situation with and and the seed polynomial . By taking successive partial derivatives, we find that an associated reduced translation-dilation invariant system of polynomials is

(2.8) |

This system has dimension , rank

(2.9) |

degree , and weight

(2.10) |

In this instance, Theorem 2.1 delivers a conclusion important enough to summarise as a corollary.

###### Corollary 2.2.

Let and be natural numbers, and let be the reduced translation-dilation invariant system given by (2.8). Suppose that is a natural number with . Then for each , one has

where .

A conclusion similar to that provided by Corollary 2.2, but with the condition replaced by

for a suitable positive constant , may be extracted from [2, Theorem 1 of Chapter III.1]. The superiority of our new bound is self-evident.

(d) Simple binary systems. A system of relevance to recent work in quantitative arithmetic geometry (see [14, §4.15],[15]) deserves to be singled out for special attention. Consider the situation with and the seed polynomial . By taking successive partial derivatives, we find that an associated reduced translation-dilation invariant system of polynomials is

(2.11) |

This system has dimension , rank

degree , and weight

By applying Theorem 2.1, we obtain the following corollary.

###### Corollary 2.3.

Let , and let be the reduced translation-dilation invariant system given by (2.11). Suppose that is a natural number with . Then for each , one has

where .

(e) The binary systems of Prendiville [11]. Consider the situation with and the seed polynomial given by the binary form of degree . In this instance, we extract the partial derivatives

and restrict attention to any subset which spans the set of all partial derivatives of positive degree, yet is linearly independent over . We take the polynomials in this spanning set to be our reduced translation-dilation invariant system . The number of partial derivatives with is plainly , while the number of monomials with is . Thus we see that this system has dimension , rank

degree and weight

In typical situations, indeed, one has . By applying Theorem 2.1, we deduce that when , one has . This conclusion may be compared with the mean value estimate underlying [11, Theorem 1.3], which delivers a similar conclusion for . The constraint on imposed in our present work is therefore stronger by a factor .

## 3. Lower bounds

In order to put into perspective the upper bounds recorded in Theorem 2.1, and such corollaries as Theorem 1.1, we consider in this section the topic of lower bounds for the mean value . Here one must consider integral solutions to the system of equations (2.1) of two types. On the one hand, there are typical solutions whose contribution to we expect to be given by a product of local densities. On the other hand, there are integral solutions lying on special subvarieties, the most obvious of which are diagonal linear spaces such as that given by . It transpires that when , there are special subvarieties not of the latter type which potentially make the dominant contribution to . In order to describe the latter subvarieties, we must introduce some further notation.

Let and be natural numbers, and consider a system of translation-dilation invariant polynomials , where . Let be a natural number with , and consider indices satisfying

(3.1) |

We say that the system of polynomials , where , is the orthogonal projection of determined by when

in which when for some index with , and when . The system remains translation-dilation invariant, and may be replaced by an equivalent reduced system . We describe as a reduced orthogonal projection of determined by . Finally, write for the set of all reduced orthogonal projections of determined by sets of indices satisfying (3.1). We remark that these orthogonal projections are in fact a special case of the more general projections introduced in the preamble to Theorem 1.7. In this section we consider only the former projections, since they are simpler to analyse and in any case deliver all of the salient features of importance for our discussion of lower bounds.

In order to facilitate our subsequent discussion, we define the polynomial by putting

(3.2) |

and then define the associated exponential sum by

(3.3) |

By orthogonality, we then have

(3.4) |

We are now equipped to describe our most general lower bound for the mean value