Random diophantine equations of additive type
aa r X i v : . [ m a t h . N T ] A p r Random diophantine equations of additive type
J. Br¨udern and R. Dietmann
I. Introduction.
In this memoir, we investigate the solubility of diagonal diophantine equations(1.1) a x k + a x k + . . . + a s x ks = 0 , and the distribution of their solutions. This is a theme that has received muchinterest in the past (see Vaughan [19], Vaughan and Wooley [20], Heath-Brown[8], Swinnerton-Dyer [16] and the extensive bibliographies in [19, 20]). Our mainconcern is with the validity of the Hasse principle, and with a bound for the smallestnon-zero solution in integers whenever such a solution exists. The approach is of astatistical nature. Very roughly speaking, we shall show that whenever s > k andthe vector a = ( a , . . . , a s ) ∈ Z s is chosen at random, then almost surely the Hasseprinciple holds for (1.1), and if there are solutions in integers, not all zero, thenthere is one with | x | ≪ | a | / ( s − k − . Here and later, we write | x | = max | x j | .We now set the scene to describe our results in precise form. To avoid trivialities,suppose throughout that k ∈ N , k ≥
2, and that a j ∈ Z \{ } . Since (1.1) has thetrivial solution x = , it will be convenient to describe the equation (1.1) as soluble over a given field if there exists a solution in that field other than the trivial one.If (1.1) is soluble over R and over Q p for all primes p , then (1.1) is called locallysoluble . We denote by C = C ( k, s ) the set of all a with a j ∈ Z \{ } for which (1.1)is locally soluble. Note that whenever (1.1) is soluble over Q , then a ∈ C . Theinverse conclusion is known as the Hasse principle for the equation (1.1). Recallthat when k = 2, then the Hasse principle holds for any s , as a special case of theHasse-Minkowski theorem.Whenever s > k , a formal use of the Hardy-Littlewood method leads one toexpect an asymptotic formula for the number ̺ a ( B ) of solutions of (1.1) in integers x j within the box | x | ≤ B . This takes the shape(1.2) ̺ a ( B ) = B s − k J a Y p χ p ( a ) + o ( B s − k ) ( B → ∞ )where(1.3) χ p ( a ) = lim h →∞ p h (1 − s ) { ≤ x j ≤ p h : a x k + . . . + a s x ks ≡ p h } is a measure for the density of the solutions of (1.1) in Q p , and similarly, J a isrelated to the surface area of the real solutions of (1.1) within the box [ − , s . Aprecise definition of J a is given in (3.7) below.As we shall see later, a condition milder than the current hypothesis s > k suffices to confirm that the limits (1.3) exist for all primes p , and that the Eulerproduct(1.4) S a = Y p χ p ( a )is absolutely convergent. Moreover, an application of Hensel’s lemma shows that χ p ( a ) is positive if and only if (1.1) is soluble in Q p . Likewise, one finds that J a ispositive if and only if (1.1) is soluble over R . It follows that (1.1) is locally solubleif and only if(1.5) J a S a > . Consequently, if (1.2) holds, then the equation (1.1) obeys the Hasse principle.The validity of (1.2), and hence of the Hasse principle for the underlying dio-phantine equations, is regarded to be a save conjecture in the range s > k , andin the special case k = 2, s > k = 3, the formula (1.2) isknown to hold whenever s ≥ s ≥ k , much less is known. The asymptoticformula (1.2) has been established when s ≥ k log k (1 + o (1)), and the Hasseprinciple may be verified when s ≥ k log k (1 + o (1)), see Ford [10] and Wooley[21]. Although these results fall short of the expected one by a factor of log k atleast, with respect to the number of variables, it seems difficult to establish (1.2)on average over a when s is significantly smaller than in the aforementioned workof Ford. However, one may choose B as a suitable function of | a | , say B = | a | θ , andthen investigate whether (1.5) holds for almost all a . This approach is successfulwhenever s > k and θ is approximately as large as 2 / ( s − k ), and suffices toconfirm the conclusions alluded to in the introductory paragraph. The principalstep is contained in the following mean value theorem. Before this is formulated,recall that a is reserved for integral vectors with non-zero entries; this conventionapplies within the summation below, and elsewhere in this paper. Also, when s isa natural number, let ˆ s denote the largest even integer strictly smaller than s . Theorem 1.1.
Let k ≥ and s > k . Then there is a positive number δ such thatwhenever A, B are real numbers satisfying (1.6) 1 ≤ B k ≤ A ≤ B (ˆ s − k ) / , one has X | a |≤ A | ̺ a ( B ) − J a S a B s − k | ≪ A s − − δ B s − k . This theorem actually remains valid when k = 2, but the proof we give belowneeds some adjustments. We have excluded k = 2 from the discussion mainlybecause in that particular case one can say more, by different methods. Hence,from now on, we assume throughout that k ≥ a with | a | ≤ A for which the inequality(1.7) | ̺ a ( B ) − J a S a B s − k | > | a | − B s − k − δ holds, does not exceed O ( A s − δ ). To deduce the Hasse principle for those a where(1.7) fails, one needs a lower bound for J a S a whenever this number is non-zero.When k is odd, (1.1) is soluble over R , and one may show that(1.8) J a ≫ | a | − holds for all a . When k is even, (1.1) is soluble over R if and only if the a j are notall of the same sign, and if this is the case, then again (1.8) holds. These facts willbe demonstrated in §
3. For the “singular product” we have the following result.
Theorem 1.2.
Let s ≥ k + 3 , and let η be a positive number. Then there exists apositive number γ such that {| a | ≤ A : 0 < S a < A − η } ≪ A s − γ . We are ready to derive the main result. Let s > k , and let δ be the positivenumber supplied by Theorem 1.1. Suppose that a ∈ C ( k, s ) satisfies A < | a | ≤ A ,and choose B = A / (ˆ s − k ) in accordance with (1.6). In Theorem 1.2, we take η = δ/ (ˆ s − k ) so that A η = B δ/ . If a is not counted in Theorem 1.2, then S a ≥ A − η , and if a also violates (1.7), then by (1.8) one has ̺ a ( B ) ≥ J a S a B s − k − | a | − B s − k − δ ≫ B s − k A − − η ≫ A − η . It follows that (1.1) has an integral solution with 0 < | x | ≤ B ≪ | a | / (ˆ s − k ) , forthese choices of a . The remaining a ∈ C ( k, s ) with A < | a | ≤ A are counted in(1.7) or in Theorem 1.2. Therefore, there are at most O ( A s − min( δ,γ ) ) such a . Wenow sum for A over powers of 2 to conclude as follows. Theorem 1.3.
Let s > k . Then, there is a positive number θ such that thenumber of a ∈ C ( k, s ) for which the equation (1.1) has no integral solution in therange < | x | ≤ | a | / (ˆ s − k ) , does not exceed O ( A s − θ ) . Browning and Dietmann [4] have recently shown that whenever s ≥
4, then(1.9) { a ∈ C ( k, s ) : | a | ≤ A } ≫ A s , so that the estimate in Theorem 1.3 is indeed a non-trivial one. In particular, itfollows that when s > k , then for almost all a ∈ C ( k, s ), the equation (1.1) issoluble over Q . Since the Hasse principle may fail for a ∈ C ( k, s ) only, this impliesthat the Hasse principle holds for almost all a ∈ C ( k, s ), but also for almost all a ∈ Z s , whenever s > k . Finally, in the same range for s , Theorem 1.3 impliesthat for almost all a for which (1.1) has non-trivial integral solutions, there existsa solution with 0 < | x | ≤ | a | / (ˆ s − k ) . This last corollary is rather remarkable, inparticular since the upper bound on the size of the solution is quite small, and nottoo far from the best possible one, as the following result shows. Theorem 1.4.
Let s > k , and let η > . Then, there exists a number c = c ( k, s, η ) > such that the number of a with | a | ≤ A for which (1.1) admits anintegral solution in the range < | x | ≤ c | a | / ( s − k ) , does not exceed ηA s . One should compare this with the lower bound (1.9): even among the locally sol-uble equations (1.1), those that have an integral solution with 0 < | x | < c | a | / ( s − k ) form a thin set, at least when c is small. It follows that the exponent 2 / (ˆ s − k )that occurs in Theorem 1.3 cannot be replaced by a number smaller than 1 / ( s − k ).An estimate for the smallest non-trivial solution of an additive diophantine equa-tion is of considerable importance in diophantine analysis, also for applications indiophantine approximation; see Schmidt [15] for a prominent example and Birch[2] for further comments. There are some bounds of this type available in the lit-erature (eg. Pitman [11]), most notably by Schmidt [14, 13]. In this context, itis worth recalling that when s > k then the equation (1.1) is soluble over Q p ,for all primes p (Davenport and Lewis [6]). When k is odd, we then expect that(1.1) is soluble over Q , and Schmidt [14] has shown that for any ε > s ( k, ε ) such that whenever s ≥ s then any equation (1.1) has an integer solutionwith 0 < | x | ≪ | a | ε . The number s ( k, ε ) is effectively computable, but Schmidt’smethod only yields poor bounds (see Hwang [9] for a discussion of this matter).When k is even and s > k , then (1.1) is locally soluble provided only that the a j are not all of the same sign. In this situation Schmidt [13] demonstrated that there still is some s ( k, ε ) such that whenever at least s ( k, ε ) of the a j are positive, andat least s ( k, ε ) are negative, then the equation (1.1) is soluble in integers with0 < | x | ≤ | a | /k + ε ;see also Schlickewei [12] when k = 2. Schmidt’s result is essentially best possible:if a ≤ b are coprime natural numbers, and k is even, then any nontrivial solution of(1.10) a ( x k + . . . + x kt ) − b ( x kt +1 + . . . + x ks ) = 0must have b | x k + . . . + x kt , whence | x | ≥ ( b/s ) /k . Thus, there are equations (1.1)where the smallest solution is as large as | a | /k , even when s is very large. However,in Theorem 1.3 the exponent 2 / (ˆ s − k ) is smaller than 1 /k . It follows that at leastwhen k is even, the exceptional set for a that is estimated in Theorem 1.3, is non-empty. On the other hand, Theorem 1.3 tell us that examples such as (1.10) wherethe smallest integer solution is large, must be sparse.We are not aware of any previous attempts to examine additive diophantineequations on average, save for the recent dissertation of Breyer [3]. There, anestimate is obtained that is roughly equivalent to a variant of Theorem 1.1 inwhich B ≍ A /k , and where the sum over a is restricted to a rather unnaturallydefined, but reasonably dense subset of Z s . In particular, Breyer’s estimates arenot of strength sufficient to derive the Hasse principle for almost all equations (1.1)with a ∈ C ( k, s ), even when s is much larger than 4 k . Yet, our analysis in section 2has certain features in common with Breyer’s work, most notably the use of latticepoint counts to treat a certain auxiliary equation. The method could be describedas an attempt to exchange the roles of coefficients and variables in (1.1). It is apure counting device, we cannot describe the exceptional sets beyond bounds ontheir cardinality. We postpone a detailed description of our methods until they areneeded in the course of the argument, but remark that the ideas developed hereincan be refined further, and may be applied to related problems as well. With morework and a different use of the geometry of numbers, we may advance into therange 3 k < s ≤ k . Perhaps more importantly, one may derive results similar tothose announced as Theorem 1.3 for the class of general forms of a given degree.Details must be deferred to sequels of this paper. Notation . Our notation is standard, or is otherwise explained within the text.Vectors are typeset in bold, and have dimension s unless indicated otherwise. Thesymbol a is reserved for tupels ( a , . . . , a s ) with non-zero integers a j . We use( x ; . . . ; x s ) to denote the greatest common divisor of the integers x j . The exponen-tial exp(2 πiα ) is abbreviated to e ( α ). Finally, we apply the familiar ε -convention:whenever ε occurs in a statement, it is asserted that the statement is valid for anypositive real number ε . Implicit constants in Landau’s or Vinogradov’s symbols areallowed to depend on ε in such circumstances. II. Applications of the geometry of numbers2.1. An elementary upper bound estimate.
Our first goal is the demonstra-tion of Theorem 1.4. The following lattice point count is the main ingredient.
Lemma 2.1.
Let c ∈ Z s be a primitive vector. Then, for any X ≥ | c | , one has { x ∈ Z s : | x | ≤ X, c x + . . . + c s x s = 0 } ≪ | c | − X s − . Proof.
See Heath-Brown [7], Lemma 1, for example.
Now let Ξ(
A, B ) denote the number of all a with | a | ≤ A for which the equation(1.1) has an integral solution with 0 < | x | ≤ B . We proceed to derive an upperbound for Ξ( A, B ). Note that whenever a is counted by Ξ( A, B ), then ̺ a ( B ) − ≥ A, B ) ≤ X | a |≤ A ( ̺ a ( B ) −
1) = X < | x |≤ B {| a | ≤ A : a x k + . . . + a s x ks = 0 } Whenever B k ≤ A , Lemma 2.1 supplies the estimateΞ( A, B ) ≪ X < | x |≤ B A s − ( x k ; . . . ; x ks ) | x | k . By symmetry, it suffices to sum over all x with x = | x | . We sort the remainingsum according to d = ( x ; x ; . . . ; x s ). Then d | x j for all j , and we infer thatΞ( A, B ) ≪ A s − X ≤ x ≤ B X d | x (cid:16) dx (cid:17) k (cid:16) X y ≤ x d | y (cid:17) s − . Since x /d ≥ d | x , it follows thatΞ( A, B ) ≪ A s − X ≤ x ≤ B X d | x (cid:16) xd (cid:17) s − − k . In particular, this confirms the following.
Lemma 2.2.
Let s ≥ k + 3 , and suppose that B k ≤ A . Then Ξ( A, B ) ≪ A s − B s − k . The proof of Theorem 1.4 is now straightforward. When s > k and 0 < C ≤ B = CA / ( s − k ) is admissible in Lemma 2.2. Let η >
0. Then, if C is sufficiently small, Lemma 2.2 supplies the inequality Ξ( A, CA / ( s − k ) ) < ηA s . If a is a vector such that | a | ≤ A and (1.1) has an integral solution with 0 < | x | The number of pairs ( x, y ) ∈ Z with | x | ≤ B , | y | ≤ B and x k ≡ y k mod d does not exceed O ( B ε + B ε d ε − /k ) .Proof. Pairs with xy = 0 contribute O ( B ). We sort the remaining pairs accordingto the value of e = ( x ; y ), and write x = ex , y = ey . The congruence implies e k | d ,and then reduces to x k ≡ y k mod de − k with 1 ≤ | x | ≤ B/e , 1 ≤ | y | ≤ B/e and( x ; y ) = 1. There are 2 B/e choices for y , and since we have now assured that( x ; de − k ) = 1, the theory of k -th power residues and a divisor function estimateyield the bound O (1 + B ε d − e k − ) for the number of choices for x , for anyadmissible choice of y . It follows that the number in question does not exceed ≪ B + B ε X e k | d d − e k − + B X e k | d e − , which confirms the lemma.Now let t be a natural number, and let V t ( A, B ) denote the number of solutionsof the equation(2.1) t X j =1 a j ( x kj − y kj ) = 0in integers a j , x j , y j constrained to(2.2) 0 < | a j | ≤ A, | x j | ≤ B, | y j | ≤ B, x kj = y kj . Lemma 2.4. Let t ≥ , and suppose that A ≥ B k ≥ . Then V t ( A, B ) ≪ A t − ( B t +1 + B t − k ) B ε . Proof. We have | x kj − y kj | ≤ B k ≤ A . Therefore, by Lemma 2.1, V t ( A, B ) ≪ A t − X | x j |≤ B X | y j |≤ Bx kj = y kj ≤ j ≤ t ( x k − y k ; . . . ; x kt − y kt )max | x kj − y kj | . By symmetry, it suffices to estimate the portion of the remaining sum where | x j | ≤ x , | y j | ≤ x for all j . Then x > 0, and we deduce that V t ( A, B ) ≪ A t − X ≤ x ≤ B X | y |≤ x y k = x k X | x j |≤ x X | y j |≤ x x kj = y kj ≤ j ≤ t ( x k − y k ; . . . ; x kt − y kt ) x k − y k . For any pair x , y with x k = y k , the inner sum will now be sorted according to thevalue of d = ( x k − y k ; . . . ; x kt − y kt ). Then d | x kj − y kj for all j = 1 , . . . , t . Therefore,by Lemma 2.3, V t ( A, B ) ≪ A t − X ≤ x ≤ B X | y |≤ x y k = x k X d | x k − y k dx k − y k X | x j |≤ x X | y j |≤ x x kj ≡ y kj mod d ≤ j ≤ t ≪ A t − X ≤ x ≤ B X | y |≤ x y k = x k X d | x k − y k dx k − y k ( x ε + x ε d − /k ) t − . Here we apply the trivial inequality ( ξ + η ) t − ≪ ξ t − + η t − that is valid fornon-negative reals ξ, η , and note that a standard divisor argument yields X ≤ x ≤ B X | y |≤ x y k = x k X d | x k − y k dx t − x k − y k ≪ B ε X ≤ x ≤ B x t ≪ B t +1+ ε so that we now deduce that(2.3) V t ( A, B ) ≪ A t − (cid:16) B t +1+ ε + B ε Υ t ( B ) (cid:17) with(2.4) Υ t ( B ) = X ≤ x ≤ B X | y |≤ xy k = x k X d | x k − y k x t − x k − y k d − t − /k . We proceed with examining two cases separately. First suppose that 2( t − ≥ k .Then, by a divisor function estimate,Υ t ( B ) ≪ B ε X ≤ x ≤ B X | y |≤ xy k = x k x t − x k − y k . When k is even, we group together the two terms ± y , and then put h = x − y . For y ≥ 0, we have x k − y k = h ( x k − + . . . + y k − ) ≥ hx k − whence Υ t ( B ) ≪ B ε X ≤ x ≤ B X ≤ y 0, are even simplerto control. Since k is odd, we have x k − y k ≥ x k , and so, X ≤ x ≤ B X − x ≤ y< x t − x k − y k ≤ X ≤ x ≤ B x t − − k ≪ B t − k + ε . It follows that Υ t ( B ) ≪ B t − k + ε holds in all cases, and by (2.3), we have nowshown that whenever 2( t − ≥ k , one has V t ( A, B ) ≪ A t − ( B t +1+ ε + B t − k + ε ) , as required.It remains to investigate the situation where 2( t − < k . Here, a divisor functionestimate applied within (2.4) yieldsΥ t ( B ) ≪ X ≤ x ≤ B x t − X | y |≤ xy k = x k ( x k − y k ) − t − /k . When k is even, we manipulate this sum much as in the previous case, and findthat Υ t ( B ) ≪ X ≤ x ≤ B x t − X ≤ y 1, apply Lemma 2.4 to bound V r ( A, B ) for 2 ≤ r ≤ t , and then deduce thefollowing estimate. Theorem 2.5. Let t ≥ . Then, for real numbers A, B with A ≥ B k ≥ , one has U t ( A, B ) ≪ ( AB ) t + A t − B t − k + ε . III. Local solubility3.1. The singular integral. Local solubility of additive equations has beeninvestigated by Davenport and Lewis [6], and by Davenport [5]. The analyticcondition (1.5) for local solubility is implicit in [6]. Unfortunately, these prominentreferences are insufficient for our purposes. A lower bound for J a S a in terms of | a | is needed whenever this product in non-zero, at least for almost all a . An estimateof this type is supplied in this section.We begin with the singular integral. Most of our work is routine, so we shall bebrief. When β ∈ R , B > 0, let(3.1) v ( β, B ) = Z B − B e ( βξ k ) dξ. A partial integration readily confirms the bound(3.2) v ( β, B ) ≪ B (1 + B k | β | ) − /k whence whenever s > k one has(3.3) Z ∞−∞ | v ( β, B ) | s dβ ≪ B s − k . We also see that for s > k and a ∈ ( Z \{ } ) s , the integral(3.4) J a ( B ) = Z ∞−∞ v ( a β, B ) . . . v ( a s β, B ) dβ converges absolutely. By H¨older’s inequality and (3.3), Z ∞−∞ | v ( a β, B ) . . . v ( a s β, B ) | dβ ≤ s Y j =1 (cid:16) Z ∞−∞ | v ( a j β, B ) | s dβ (cid:17) /s ≪ | a . . . a s | − /s B s − k . (3.5) In particular, it follows that(3.6) J a ( B ) ≪ | a . . . a s | − /s B s − k . The integral J a ( B ) arises naturally as the singular integral in our application ofthe circle method in section 4. The dependence on B can be made more explicit.By (3.1), one has v ( β, B ) = Bv ( βB k , β for βB k in (3.4) to inferthat(3.7) J a ( B ) = B s − k J a where J a = J a (1) is the number that occurs in (1.2), and in Theorem 1.1.It remains to establish a lower bound for J a . The argument depends on theparity of k , and we shall begin with the case when k is even. Throughout, wesuppose that(3.8) | a s | ≥ | a j | (1 ≤ j < s ) . Define σ j = a j / | a j | ∈ { , − } . Then, by (3.1), v ( a j β, 1) = 2 Z e ( a j βξ k ) dξ = 2 k | a j | − /k Z | a j | η (1 − k ) /k e ( σ j βη ) dη. Let A = [0 , | a | ] × . . . × [0 , | a s | ], and define the linear form τ through the equation(3.9) σ s τ = σ η + . . . + σ s η s . Then, we may rewrite (3.4) as J a = (cid:16) k (cid:17) s | a . . . a s | − /k Z ∞−∞ Z A ( η . . . η s ) (1 − k ) /k e ( σ s τ β ) d η dβ. Now substitute τ for η s in the innermost integral. Then, by Fubini’s theorem and(3.9), Z A ( η . . . η s ) (1 − k ) /k e ( σ s τ β ) d η = Z ∞−∞ E ( τ ) e ( σ s τ β ) dτ where(3.10) E ( τ ) = Z E ( τ ) ( η . . . η s − η s ( τ, η , . . . , η s − )) (1 − k ) /k d ( η , . . . , η s − ) , in which η s is the linear form defined implicitly by (3.9), and where E ( τ ) is the setof all ( η , . . . , η s − ) satisfying the inequalities0 ≤ η j ≤ | a j | (1 ≤ j < s ) , ≤ τ − σ s σ η − σ s σ η − . . . − σ s σ s − η s − ≤ | a s | . It transpires that E is a non-negative continuous function with compact support,and that for τ near 0, this function is of bounded variation. Therefore, by Fourier’sintegral theorem, lim N →∞ Z N − N Z ∞−∞ E ( τ ) e ( σ s τ β ) dτ dβ = E (0) , and we infer that(3.11) J a = (cid:16) k (cid:17) s | a . . . a s | − /k E (0) . In particular, it follows that J a ≥ 0. Also, when all a j have the same sign, then E (0) = { } , and (3.11) yields J a = 0. Now suppose that not all the a j are of the same sign. First, consider the situationwhere σ = . . . = σ s − . Then we have σ s σ j = − ≤ j < s ). By (3.8), we seethat the set of ( η , . . . , η s − ) defined by | a j | s ≤ η j ≤ | a j | s (1 ≤ j < s )is contained in E (0), and its measure is bounded below by (2 s ) − s | a a . . . a s − | . By(3.10), we now deduce that E (0) ≫ | a . . . a s − | /k | a s | (1 − k ) /k , and (3.11) then implies the bound J a ≫ | a s | − = | a | − .In the remaining cases, both signs occur among σ , . . . , σ s − . We may thereforesuppose that for some r with 2 ≤ r < s we have σ s σ j = − ≤ j < r ) , σ s σ j = 1 ( r ≤ j < s ) . Take τ = 0 in (3.9). Then η s is the linear form(3.12) η s = η + . . . + η r − − η r − . . . − η s − . By symmetry, we may suppose that | a | ≤ | a | ≤ . . . ≤ | a r − | , | a r | ≤ | a r +1 | ≤ . . . ≤ | a s − | . We define t by t = r − | a r − | ≤ | a r | , and otherwise as the largest t among r, r + 1 , . . . , s − | a t | ≤ | a r − | . Now consider the set of ( η , . . . , η s − ) definedby the inequalities | a j | s ≤ η j ≤ | a j | s (1 ≤ j ≤ r − , | a j | s ≤ η j ≤ | a j | s ( r ≤ j ≤ t ) , | a r − | s ≤ η j ≤ | a r − | s ( t < j < s ) . It is readily checked that on this set, the number η s defined in (3.12) satisfiesthe inequalities | a r − | s ≤ η s ≤ | a r − | . Moreover, the measure of this set is ≫| a . . . a t || a r − | s − t +2 . By (3.10), it follows that E (0) ≫ | a . . . a t | /k | a r − | ( s − t +2) /k | a r − | (1 − k ) /k , and again one then deduces from (3.11) the bound J a ≫ | a | − .Finally, we discuss the case where k is odd. The main differences in the treatmentoccur in the initial steps. When k is odd, one may transform (3.1) into v ( a j β, 1) = 1 k | a j | − /k Z | a j | η (1 − k ) /k ( e ( βη ) + e ( − βη )) dη. Let σ = ( σ , . . . , σ s ) with σ j ∈ { , − } . For any such σ , define τ through (3.9).Then, following through the argument used in the even case, we first arrive at theidentity J a = k − s | a . . . a s | − /k X σ Z ∞−∞ Z A ( η . . . η s ) (1 − k ) /k e ( σ s τ β ) d η dβ. Here the sum is over all 2 s choices of σ . Again as before, we see that each individualsummand is non-negative, and when not all of σ , . . . , σ s have the same sign, then one finds the lower bound ≫ | a | − for this summand. Thus, we now see that J a ≫ | a | − again holds, this time for any choice of a .For easy reference, we summarize the above results as a lemma. Lemma 3.1. Suppose that s > k . Then the singular integral J a converges absolutely,and one has ≤ J a ≪ | a a . . . a s | /s . Furthermore, when k is odd, or when k iseven and a , . . . , a s are not all of the same sign, then J a ≫ | a | − . Otherwise J a = 0 . In the introduction, we defined the classical singularseries as a product of local densities. We briefly recall its representation as a series.Though this is standard in principle, our exposition makes the dependence on thecoefficients a in (1.1) as explicit as is necessary for the proof of Theorem 1.2 in thenext section. Recall that k ≥ q ∈ N , r ∈ Z define the Gaussian sum(3.13) S ( q, r ) = q X x =1 e ( rx k /q ) . Let κ ( q ) be the multiplicative function that, on prime powers q = p l , is given by κ ( p uk + v ) = p − u − ( u ≥ , ≤ v ≤ k ) , κ ( p uk +1 ) = kp − u − / . Then, as a corollary to Lemmas 4.3 and 4.4 of Vaughan [19], one has S ( q, r ) ≪ qκ ( q )whenever ( q ; r ) = 1, and one concludes that(3.14) q − S ( q, r ) ≪ κ ( q/ ( q ; r ))holds for all q ∈ N , r ∈ Z . Now let(3.15) T a ( q ) = q − s q X r =1( r ; q )=1 S ( q, a r ) . . . S ( q, a s r ) . Then, by (3.14),(3.16) T a ( q ) ≪ qκ ( q/ ( q ; a )) . . . κ ( q/ ( q ; a s )) . Moreover, by working along the proof of Lemma 2.11 of Vaughan [19], one findsthat T a ( q ) is a multiplicative function of q . Also, one can use the definition of κ to confirm that whenever s ≥ k + 2 then the expression on the right hand side of(3.16) may be summed over q to an absolutely convergent series. Thus, we mayalso sum T a ( q ) over q and rewrite the series as an Euler product. This gives(3.17) ∞ X q =1 T a ( q ) = Y p ∞ X h =0 T a ( p h ) . However, by (3.13) and (3.15), and orthogonality,(3.18) l X h =0 T a ( p h ) = p − ls p l X r =1 S ( p l , ra ) . . . S ( p l , ra s ) = p l (1 − s ) M a ( p l )where M a ( p l ) is the number of incongruent solutions of the congruence a x k + . . . + a s x ks ≡ p l . We may take the limit for l → ∞ in (3.18) because all sums in (3.17) are convergent.This shows that the limit χ p , as defined in (1.3), exists. In view of (3.17) and (1.4), we may summarize our results as follows. Lemma 3.2. Let s ≥ k + 2 . Then, for any a ∈ ( Z \{ } ) s , the singular product (1.4) converges, and has the alternative representation S a = ∞ X q =1 T a ( q ) . A slight variant of the preceding argument also supplies an estimate for χ p ( a )when p is large. Lemma 3.3. Let s ≥ k + 2 . Then there is a real number c = c ( k, s ) such that forany choice of a , . . . , a s ∈ Z \{ } for which at least k + 2 of the a j are not divisibleby p , one has | χ p ( a ) − | ≤ cp − . Proof. We begin with (3.18), and note that T a (1) = 1. Then p l (1 − s ) M a ( p l ) − l X h =1 T a ( p h ) . One has κ ( q ) ≤ k for any prime power q . Hence, by (3.16), and since k + 2 of the a j are coprime to p , one finds that | T a ( p h ) | ≤ k s κ ( p h ) k +2 p h . Consequently, a shortcalculation based on the definition of κ reveals that | p l (1 − s ) M a ( p l ) − | ≤ k s l X h =1 κ ( p h ) k +2 p h ≤ k s + k +2 p − . The lemma follows on considering the limit l → ∞ . Throughout, we suppose that s ≥ k + 3. For a ∈ ( Z \{ } ) s , let S ( a ) denote the set of all primes that divide at least two of theintegers a j . Lemma 3.3 may then be applied to all primes p / ∈ S ( a ), and we deducethat there exists a number C = C ( k, s ) > ≤ Y p/ ∈S ( a ) p>C χ p ( a ) ≤ a . It will be convenient to write P ( a ) = S ( a ) ∪ { p : p ≤ C } ;this set contains all primes not covered by (3.19). For a prime p ∈ P ( a ), let l ( p ) = max { l : p l | a j for some j } , and then define the numbers P ( a ) = Y p ∈P ( a ) p, P ( a ) = Y p ∈S ( a ) p>C p, P † ( a ) = Y p ∈P ( a ) p l ( p ) . For later use, we note that P ( a ) | P ( a ) , P ( a ) | HP ( a )in which we wrote H = Y p ≤ C p. Now fix a number δ > 0, to be determined later, and consider the sets A = {| a | ≤ A : P ( a ) > A δ } , (3.20) A = {| a | ≤ A : P ( a ) ≤ A δ , P † ( a ) > A δ } (3.21)It transpires that the set A ∪ A contains all a where the singular series is likely tobe smallish. Fortunately, A and A are defined by divisibility constraints that arerelated to convergent sieves, so one expects A , A to be thin sets. This is indeedthe case, as we shall now show.We begin by counting elements of A . For a natural number d , let A ( d ) = { a ∈A : P ( a ) = d } . If there is some a ∈ A ( d ), then by the definition of S ( a ), we have d | a a . . . a s , whence d ≤ A s/ . On the other hand, A δ < P ( a ) ≤ HP ( a ) ≤ Hd .This shows that A = X A δ /H Let ν ≥ be a real number. Then, { n ≤ X ν : n ∗ ≤ X } ≪ X ε . For d ∈ N , let A ( d ) = { a ∈ A : P † ( a ) = d } . Since we have P † ( a ) | a a . . . a s , wemust have A δ < d ≤ A s whenever A ( d ) is non-empty. Moreover, P ( a ) is the square-free kernel of P † ( a ),so that d ∗ ≤ P δ . This yields the bound A = X A δ 0, whence (1.1) is soluble in Q p . By homogeneity, there is then a solution x ∈ Z p of (1.1) with p ∤ x . In particular, for any h ∈ N , we can find integers y , . . . , y s that are not all divisible by p , and satisfy the congruence(3.24) a y k + . . . + a s y ks ≡ p h . It will be convenient to rearrange indices to assure that p ∤ y . Let ν ( p ) be definedby p ν ( p ) k k , and recall that a k -th power residue mod p ν ( p )+2 is also a k -th powerresidue modulo p ν , for any ν ≥ ν ( p )+2. We choose h = l ( p )+ ν ( p )+2 in (3.24), anddefine e by p e k a . For l > h , choose numbers x j , for 2 ≤ j ≤ s , with 1 ≤ x j ≤ p l and x j ≡ y j mod p h . Then, by (3.24), − a p e y k ≡ a x k + . . . + a s x ks p e mod p h − e , and we have e ≤ l ( p ), whence h − e ≥ ν ( p ) + 2. Thus, for any choice of x , . . . , x s as above, there is a number x with a x k + . . . + a s x ks ≡ p l . Counting the number of possibilities for x , . . . , x s yields M a ( p l ) ≥ p ( s − l − h ) , andconsequently, χ p ( a ) ≥ p (1 − s ) h . We may combine this with (3.19) to infer that(3.25) S a ≥ Y p ∈P ( a ) p (1 − s ) h . In this product, we first consider primes p ∈ P ( a ) where l ( p ) = 0. Then p ∤ a a . . . a s , and the definition of P ( a ) implies that p ≤ C . Also, since ν ( p ) ≤ k , wehave h ≤ k + 2 so that Y p ∈P ( a ) l ( p )=0 p (1 − s ) h ≥ Y p ≤ C p (1 − s )( k +2) ≥ H (1 − s )( k +2) . Next, consider p ∈ P ( a ) with l ( p ) ≥ 1. Then, much as before, h ≤ k + 2 + l ( p ) ≤ l ( p )( k + 3). Hence, Y p ∈P ( a ) l ( p ) ≥ p (1 − s ) h ≥ P † ( a ) (1 − s )( k +3) . However, since a / ∈ A ∪ A , we have P † ( a ) ≤ A δ , so that we now deduce from(3.25) that(3.26) S a ≫ A δ (1 − s )( k +3) . The synthesis is straightforward. Let γ > 0. Then choose δ = γ/ (8( s − k + 3)),and suppose that A is large. Then (3.26) implies that S a > A − γ . If that fails, then S a = 0, or else a ∈ A ∪ A . The estimates (3.22) and (3.23) imply Theorem 1.2. We close this section with a succession oflemmata that involve the function κ , and that will provide an upper bound for S a on average. The results will be relevant for the application of the circle method inthe next section. Lemma 3.5. One has X d | q dκ ( d ) ≪ q ε κ ( q ) . Proof. Let p be a prime, and suppose that 0 ≤ j ≤ l . Then, an inspection of thedefinition of κ readily reveals that the crude inequality p j κ ( p j ) ≤ kp l κ ( p l ) holds.Consequently, one also has X d | p l dκ ( d ) = l X j =1 p j κ ( p j ) ≤ k ( l + 1) κ ( p l ) p l . By multiplicativity, this implies the bound X d | q dκ ( d ) ≤ qκ ( q ) Y p l k q k ( l + 1) , which is more than required. Lemma 3.6. Uniformly for q ∈ N and A ≥ , one has X ≤ a ≤ A κ ( q/ ( q ; a )) ≪ Aq ε κ ( q ) . Proof. We sort the a according to the value of d = ( q ; a ). Then X ≤ a ≤ A κ ( q/ ( q ; a )) = X d | qd ≤ A κ ( q/d ) X ≤ a ≤ A ( a ; q )= d ≤ A X d | q d − κ ( q/d ) = Aq X d | q dκ ( d ) . The lemma now follows by appeal to Lemma 3.5. Lemma 3.7. Let s ≥ k + 2 . Then X | a |≤ A ∞ X q =1 ( a a . . . a s ) − /s q / (2 k ) κ ( q/ ( q ; a )) . . . κ ( q/ ( q ; a s )) ≪ A s − . Proof. The terms to be summed are non-negative. Thus, we may take the sumover a first. This then factorizes, and by Lemma 3.6 and partial summation, theleft hand side in Lemma 3.7 is seen not to exceed A s − ∞ X q =1 q / (2 k )+ ε κ ( q ) s . The remaining sum converges for s ≥ k + 2, as one readily confirms by consideringthe corresponding Euler product. The lemma follows.We now apply the last estimate to the singular series. Let T a ( q ) be as in (3.15).When Q ≥ 1, define the tail of S a as(3.27) S a ( Q ) = X q ≥ Q T a ( q )which is certainly convergent for s ≥ k + 2; compare Lemma 3.2. Also, note that S a = S a (1). Lemma 3.8. Let s ≥ k + 2 . Then, uniformly in A ≥ , Q ≥ , one has X | a |≤ A ( a a . . . a s ) − /s | S a ( Q ) | ≪ A s − Q − / (2 k ) . Proof. By (3.16), | S a ( Q ) | ≤ Q − / (2 k ) ∞ X q =1 q / (2 k ) | T a ( q ) |≤ Q − / (2 k ) ∞ X q =1 q / (2 k ) κ ( q/ ( q ; a )) . . . κ ( q/ ( q ; a s )) , and the lemma follows from Lemma 3.7. IV. The circle method4.1. Preparatory steps. In this section, we establish Theorem 1.1. Theargument is largely standard, save for the ingredients to be imported from theprevious sections of this memoir.We employ the following notational convention throughout this section: if h : R → C is a function, and a ∈ Z s , then we define(4.1) h a ( α ) = h ( a α ) h ( a α ) . . . h ( a s α ) . As is common in problems of an additive nature, the Weyl sum(4.2) f ( α ) = X | x |≤ B e ( αx k )is prominently featured in the argument to follow, because by orthogonality, onehas(4.3) ̺ a ( B ) = Z f a ( α ) dα. The circle method will be applied to the integral in (4.3). With applications inmind that go well beyond those in the current communication, we shall treat the“major arcs” under very mild conditions on A, B , and for the range s ≥ k + 2.Let A ≥ B ≥ 1, and fix a real number η > 0. Then put Q = B η . Let M denote the union of the intervals(4.4) (cid:12)(cid:12)(cid:12) α − rq (cid:12)(cid:12)(cid:12) ≤ QAB k with 1 ≤ r ≤ q < Q , and ( r, q ) = 1. When η ≤ , these intervals are pairwisedisjoint, and we write m = [ Q/ ( AB k ) , Q/ ( AB k )] \ M . When A is one of M or m , let(4.5) ̺ a ( B, A ) = Z A f a ( α ) dα and note that(4.6) ̺ a ( B ) = ̺ a ( B, M ) + ̺ a ( B, m ) . In this section we make heavy use of the resultsin Vaughan’s book [19] on the subject. He works with the Weyl sum g ( α ) = X ≤ x ≤ B e ( αx k )that is related with our f through the formulae f ( α ) = 1 + 2 g ( α ) ( k even) , f ( α ) = 1 + g ( α ) + g ( − α ) ( k odd) . Thus, in particular, Theorem 4.1 of [19] yields the following. Lemma 4.1. Let α ∈ R , r ∈ Z , q ∈ N and a ∈ Z with a = 0 . Then f ( aα ) = q − S ( q, ar ) v ( a ( α − r/q )) + O ( q / ε (1 + | a | B k | α − r/q | ) / ) . Here, and throughout the rest of this section, we define v ( β ) = v ( β, B ) through(3.1). When | a | ≤ A , and α ∈ M is in the interval (4.4), we find that f ( aα ) = q − S ( q, ar ) v ( a ( α − r/q )) + O ( Q ) . This we use with a = a j and multiply together. Then f a ( α ) = q − s S ( q, a r ) . . . S ( q, a s r ) v a ( α − r/q ) + O ( Q B s − ) . Now integrate over M , and recall the definition of the latter. By (4.5) and (3.15),we then arrive at ̺ a ( B, M ) = X q Let A ≥ , B ≥ and Q = B / . Then, whenever s ≥ k + 2 , one has X | a |≤ A | ̺ a ( B, M ) − S a J a B s − k | ≪ A s − B s − k − / (12 k ) . We begin the endgame with a variant of Weyl’s inequality. Lemma 4.3. Let A ≥ , B ≥ , and suppose that r ∈ Z and q ∈ N are coprime with | α − ( r/q ) | ≤ q − . Then X < | a |≤ A | f ( aα ) | k − ≪ AB k − (cid:16) q + 1 B + qAB k (cid:17) ( ABq ) ε . This is well known, but we give a brief sketch for completeness. Write K = 2 k − .Then, as an intermediate step towards the ordinary form of Weyl’s inequality, onehas | f ( β ) | K ≪ B K − + B K − k + ε X ≤ h ≤ k k ! B k − min( B, k hβ k − )where k β k denotes the distance of β to the nearest integer; compare the argumentsunderpinning Lemma 2.4 of Vaughan [19]. Now choose β = aα and sum over a . Adivisor function argument then yields X < | a |≤ A | f ( aα ) | K ≪ AB K − + B K − k ( AB ) ε X h ≪ AB k − min( B, k hβ k − ) , and Lemma 4.3 follows from Lemma 2.2 of Vaughan [19].Now let α ∈ m . By Dirichlet’s theorem on diophantine approximations, thereare r ∈ Z , q ∈ N with q ≤ Q − AB k and | qα − r | ≤ Q ( AB k ) − . But α / ∈ M , whence q > Q . Lemma 4.3 in conjunction with H¨older’s inequalitynow yields(4.10) sup α ∈ m X < | a |≤ A | f ( aα ) | ≪ ( AB ) ε Q − − k . We now apply this estimate to establish the following. Lemma 4.4. Let s ∈ N , s = 2 t + u with t ∈ N , u = 1 or . Then there is a number δ > such that whenever ≤ B k ≤ A ≤ B t − k holds, then X | a |≤ A | ̺ a ( B, m ) | ≪ A s − B s − k − δ . Proof. By (4.5), one has X | a |≤ A | ̺ a ( B, m ) | ≤ Z m (cid:16) X < | a |≤ A | f ( aα ) | (cid:17) s dα. Moreover, by Cauchy’s inequality and orthogonality, Z (cid:16) X < | a |≤ A | f ( aα ) | (cid:17) t dα ≤ (2 A + 1) t Z (cid:16) X < | a |≤ A | f ( aα ) | (cid:17) t dα ≤ (2 A + 1) t U t ( A, B ) . On combining the last two inequalities with (4.10) and Theorem 2.5, we deducethat(4.11) X | a |≤ A | ̺ a ( B, m ) | ≪ A s B t + u − δ + A s − B s − k − δ where any 0 < δ < − k is admissible. Note that the condition that B k ≤ A isrequired in Theorem 2.5, whereas the inequality A ≤ B t − k makes the second termon the right of (4.11) the dominating one. This establishes the lemma.Theorem 1.1 is also available: one has ˆ s = 2 t , and the theorem follows oncombining (4.6) with Lemma 4.2 and Lemma 4.4. References [1] Baker, R. C. Diagonal cubic equations. II. Acta Arith. 53 (1989), 217–250.[2] Birch, B. J. Small zeros of diagonal forms of odd degree in many variables. Proc. LondonMath. Soc. (3) 21 (1970), 12–18.[3] Breyer, T. ¨Uber Hasseprinzipien von Diagonalformen. Dissertation, Universit¨at Stuttgart.Shaker, Aachen, 2004.[4] Browning, T. D.; Dietmann, R. Solubility of Fermat equations. Quadratic forms—algebra,arithmetic, and geometry, 99–106, Contemp. Math., 493, Amer. Math. Soc., Providence, RI,2009.[5] Davenport, H. Analytic methods for Diophantine equations and Diophantine inequalities.Second edition. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman.Edited and prepared for publication by T. D. Browning. Cambridge Mathematical Library.Cambridge University Press, Cambridge, 2005.[6] Davenport, H.; Lewis, D. J. Homogeneous additive equations. Proc. Roy. Soc. Ser. A, 274(1963), 443–460.[7] Heath-Brown, D. R. The density of rational points on curves and surfaces. Ann. of Math. (2)155 (2002), 553–595.[8] Heath-Brown, D.R. The solubility of diagonal cubic diophantine equations. Proc. LondonMath. Soc. (3) 79 (1999), 241–259.[9] Hwang, J. S. Small zeros of additive forms in several variables. Acta Math. Sinica (N.S.) 14(1998), 57–66.[10] Ford, Kevin B. New estimates for mean values of Weyl sums. Internat. Math. Res. Notices1995, 155–171.[11] Pitman, J. Bounds for solutions of diagonal equations. Acta Arith. 19 (1971), 223–247.[12] Schlickewei, H. P. Kleine Nullstellen homogener quadratischer Gleichungen. Monatsh. Math.100 (1985), 35–45. [13] Schmidt, Wolfgang M. Small zeros of additive forms in many variables. Trans. Amer. Math.Soc. 248 (1979), 121–133.[14] Schmidt, W. M. Small zeros of additive forms in many variables. II. Acta Math. 143 (1979),219–232.[15] Schmidt, W. M. Diophantine inequalities for forms of odd degree. Adv. in Math. 38 (1980),128–151.[16] Swinnerton-Dyer, P. The solubility of diagonal cubic surfaces. Ann. Sci. ´Ecole Norm. Sup.(4) 34 (2001), 891–912.[17] Tenenbaum, G. Introduction to analytic and probabilistic number theory. Cambridge Uni-versity Press, Cambridge, 1995.[18] Vaughan, R. C. On Waring’s problem for cubes. J. Reine Angew. Math. 365 (1986), 122–170.[19] Vaughan, R. C. The Hardy-Littlewood method. Second edition. Cambridge Tracts in Math-ematics, 125. Cambridge University Press, Cambridge, 1997[20] Vaughan, R. C.; Wooley, T. D. Waring’s problem: a survey. Number theory for the millen-nium, III (Urbana, IL, 2000), 301–340, A K Peters, Natick, MA, 2002.[21] Wooley, T. D. Large improvements in Waring’s problem. Ann. of Math. (2) 135 (1992), 131–164.[13] Schmidt, Wolfgang M. Small zeros of additive forms in many variables. Trans. Amer. Math.Soc. 248 (1979), 121–133.[14] Schmidt, W. M. Small zeros of additive forms in many variables. II. Acta Math. 143 (1979),219–232.[15] Schmidt, W. M. Diophantine inequalities for forms of odd degree. Adv. in Math. 38 (1980),128–151.[16] Swinnerton-Dyer, P. The solubility of diagonal cubic surfaces. Ann. Sci. ´Ecole Norm. Sup.(4) 34 (2001), 891–912.[17] Tenenbaum, G. Introduction to analytic and probabilistic number theory. Cambridge Uni-versity Press, Cambridge, 1995.[18] Vaughan, R. C. On Waring’s problem for cubes. J. Reine Angew. Math. 365 (1986), 122–170.[19] Vaughan, R. C. The Hardy-Littlewood method. Second edition. Cambridge Tracts in Math-ematics, 125. Cambridge University Press, Cambridge, 1997[20] Vaughan, R. C.; Wooley, T. D. Waring’s problem: a survey. Number theory for the millen-nium, III (Urbana, IL, 2000), 301–340, A K Peters, Natick, MA, 2002.[21] Wooley, T. D. Large improvements in Waring’s problem. Ann. of Math. (2) 135 (1992), 131–164.