Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy-Littlewood circle method to first study an asymptotic version of Waring's problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov's mean value theorem, due to Bourgain-Demeter-Guth and Wooley.


Introduction
This paper is about the arithmetic of rational points on higher-dimensional orbifolds, in the spirit of Campana [4]. We shall be concerned with orbifolds (P n−1 , ∆), where ∆ is a Q-divisor that takes the shape for irreducible divisors D 0 , . . . , D r on P n−1 and integers m 0 , . . . , m r 2. The arithmetic of Campana-points on orbifolds interpolates between the theory of rational and integral points on classical algebraic varieties, thereby opening up a new field of enquiry.
The orbifold (P n−1 , ∆) is smooth if the divisor r i=0 D i is strict normal crossings and it is said to be log-Fano if −K P n−1 ,∆ is ample, where K P n−1 ,∆ = K P n−1 + ∆. Forthcoming work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado [9] introduces the notion of Campana-points on higher-dimensional orbifolds and studies their distribution on vector group compactifications. Motivated by the Manin conjecture for rational points on Fano varieties [6], it is very natural to ask what one can say about the density of Campana-points of bounded height on smooth log-Fano orbifolds (P n−1 , ∆). We shall address this in the special case that D 0 , . . . , D r form a set of distinct hyperplanes in P n−1 , all defined over Q. Then (P n−1 , ∆) is log-Fano precisely when Since m i 2 this forces us to have r 2(n − 1). It turns out that the analysis is easy when r n − 1 and so the first challenging case is when r = n, in which case the condition for being log-Fano is n i=0 1 m i > 1.
We shall take if 0 i n − 1, {c 0 x 0 + · · · + c n−1 x n−1 = 0} if i = n, for a fixed choice of non-zero integers c 0 , . . . , c n−1 . We let for given integers m i 2. The Campana-points in (P n−1 , ∆) are defined to be the rational points (x 0 : · · · : x n−1 ) ∈ P n−1 (Q), represented by primitive integer vectors (x 0 , . . . , x n−1 ) ∈ Z n =0 for which x i is m i -full for 0 i n − 1 and c 0 x 0 + · · · + c n−1 x n−1 is m n -full. Here, we recall that a non-zero integer x is said to be m-full if p m | x whenever there is a prime p such that p | x.
We attach the usual exponential height function H : P n−1 (Q) → R, given by H(x 0 : · · · : x n−1 ) = max 0 i n−1 |x i | if (x 0 , . . . , x n−1 ) ∈ Z n is primitive. The counting function of interest to us here is then gcd(x 0 , . . . , x n−1 ) = 1 |x| B, x i is m i -full ∀ i c 0 x 0 + · · · + c n−1 x n−1 = x n    , (1.1) where x = (x 0 , . . . , x n ) and |x| = max 0 i n |x i |. In the special case m 0 = · · · = m n = 2, work of Van Valckenborgh [12] establishes an asymptotic formula for N(P n−1 , ∆; B) for all n 4. Drawing inspiration from this, we have the following generalisation. The implied constant in this estimate is allowed to depend on m 0 , . . . , m n , n and c 0 , . . . , c n−1 , a convention that we shall adopt for all of the implied constants in this paper. There is an explicit expression for the leading constant c in (3.14) and (3.15), as a convergent sum of local densities. It can be shown that c > 0 if the underlying equations admit suitable non-singular solutions everywhere locally. In Theorem 1.1 the exponent of B is equal to a = a(L, ∆) = inf ℓ ∈ R : ℓ[L] + [K P n−1 ,∆ ] ∈ C eff (P n−1 ) , where [L] is the class of a hyperplane section in P n−1 . Moreover, the exponent of log B is b − 1, where b = b(L, ∆) is the codimension of the minimal face of ∂C eff (P n−1 ) that contains a[L] + [K P n−1 ,∆ ]. When m 0 = · · · = m n = 2 and n 4 the work of Van Valckenborgh [12] shows that the asymptotic formula for N(P n−1 , ∆; B) follows the same pattern, with a = n−1 2 and b = 1. However, some caution must be exercised when asking to what extent other orbifolds conform to this behaviour, as the following result shows. On the other hand, when n = 2 we expect the counting function to satisfy an asymptotic formula with associated constants a = 1 2 and b = 1. In fact, Browning and Van Valckenborgh [2] have produced an explicit constant c > 0 such that N( Let X be an integral variety over Q. Recall from Serre [11, §3.1] that a thin set is a set contained in a finite union of thin subsets of X(Q) of type I and II. Here, a type I thin subset is a set of the form Z(Q) ⊂ X(Q), where Z is a proper closed subvariety, and a type II thin subset is a set of the form f (Y (Q)), where f : Y → X is a generically finite dominant morphism with dim Y = dim X, deg f 2 and Y geometrically integral. It follows from work of Cohen [5] (as further expounded by Serre [10,Thm. 13.3]) that the set P n−1 (Q) is not thin. At the workshop "Rational and integral points via analytic and geometric methods" in Oaxaca (May 27th-June 1st, 2018), Sho Tanimoto raised the question of whether the same is true for the set of Campana-points. Our next goal is to provide some partial evidence in favour of this.
Associated to any type II thin subset Ω ⊂ P n−1 (Q) coming from a morphism Y → P n−1 of degree d 2 is a degree d extension of function fields Q(Y )/Q(t 1 , . . . , t n−1 ). We let Q(Y ) Gal be the Galois closure of Q(Y ) over the function field Q(t 1 , . . . , t n−1 ) of P n−1 and we let Q Ω ⊂ Q(Y ) Gal be the largest subfield that is algebraic over Q. Finally we let P Ω be the set of rational primes that split completely in Q Ω . It follows from the Chebotarev density theorem that P Ω has density [Q Ω : Q] −1 in the set of primes, since Q Ω /Q is Galois. Next, let for any m = (m 0 , . . . , m n ) ∈ Z n+1 2 . The following result provides an explicit condition on the possible thin sets that the Campana-points in (P n−1 , ∆) are allowed to lie in.
Assuming that (1.2) holds, we may combine this result with Theorem 1.1 to deduce that the Campana-points in (P n−1 , ∆) are not contained in any thin set satisfying the hypotheses of the theorem. The statement of this result is rather disappointing at first glance, but in fact the conclusion is false when the condition (1.4) is dropped. To see this, take m 0 = · · · = m n = 3 and n 12. Then n i=0 1 m i − 1 = n−2 3 and (1.2) holds in Theorem 1.1. Consider the thin set Ω 0 ⊂ P n−1 (Q) that arises from the morphism Z → P n−1 , (x 0 : · · · : x n ) → (x 0 : · · · : x n−1 ), where Z ⊂ P n is the cubic hypersurface x 3 0 + · · · + x 3 n−1 = x 3 n . Then the counting function N Ω 0 (P n−1 , ∆; B) has exact order B n− 2 3 for sufficiently large n. However, (1.4) fails in this case. Indeed, Q m is the set of primes p ≡ 1 mod 3, whereas P Ω 0 is the set of primes p ≡ 1 mod 3, since Q Ω 0 = Q( √ −3). This shows that it is hard to approach Tanimoto's question in full generality through counting arguments alone.
The hypothesis (1.4) is a little awkward to work with. If one restricts attention to m such that gcd(m j , m j ′ ) = 1 for 0 j < j ′ n, (1.5) then Q m is equal to the full set of rational primes. Moreover, it follows from Chebotarev's density theorem that P Ω has density [Q Ω : Q] −1 , for any type II thin subset Ω. Thus the conditions of Theorem 1.3 are met for any thin set. However, the assumption (1.2) is too stringent to cope with a sequence of integers 2 that satisfy (1.5).
Our proof of Theorems 1.1-1.3 relies on an explicit description of m-full integers x. For such integers every exponent of a prime appearing in the prime factorisation of x can be written km + (m + r), for integers k 0 and 0 r < m. Thus any non-zero m-full integer x can be written uniquely in the form (1.6) for u, v 1 , . . . , v m−1 ∈ N, such that µ 2 (v r ) = 1 for 1 r m − 1 and gcd(v r , v r ′ ) = 1 for 1 r < r ′ m − 1.
It may be instructive to illustrate this notation by discussing the special case m 0 = · · · = m n = 2, in which case Campana-points in (P n−1 , ∆) correspond to vectors u, v ∈ N n+1 and ǫ ∈ {±1} n+1 with each v j square-free, for which ǫ 0 c 0 u 2 0 v 3 0 + · · · + ǫ n−1 c n−1 u 2 n−1 v 3 n−1 = ǫ n u 2 n v 3 n . When n = 3 we can clearly find vectors v ∈ N 4 with square-free components and ǫ ∈ {±1} 4 in such a way that Fixing such a choice and applying [7,Thm. 7] to estimate the residual number of u ∈ N 4 that lie on the split quadric, with u j B/v 3 j for 0 j 3, we readily deduce that N(P 2 , ∆; B) ≫ B log B, as claimed in Theorem 1.2 Returning now to the case of general m 0 , . . . , m n 2, we summarise the structure of the paper. Under the representation (1.6) it follows that Campanapoints on (P n−1 , ∆) can be viewed through the lens of Waring's problem for mixed exponents. Given its proximity to Vinogradov's mean value theorem, this is an area that has received a radical new injection of ideas at the hands of Wooley [14,16,17] and Bourgain, Demeter and Guth [1]. Based on this, in §2 we shall give a completely general treatment of the counting function associated to suitably constrained integer solutions to the Diophantine equation for given N ∈ Z and non-zero c j , γ j ∈ Z, in which the vectors u are asked to lie in a congruence class modulo H. In this part of the argument we shall need to retain uniformity in the coefficients γ j and in the modulus H. It is here that the condition (1.2) arises. The resulting asymptotic formula is recorded in Theorem 2.7. In §3 we shall use Theorem 2.7 to establish the version of orbifold Manin that we have presented in Theorem 1.1. One of the chief difficulties in this part of the argument comes from dealing with the coprimality conditions implicit in the counting function N(P n−1 , ∆; B). Next, in §4 we shall combine Theorem 2.7 with information about the size of thin sets modulo p (for many primes p) to tackle Theorem 1.3.
Finally, when H = 1 and c j = γ j = 1 for all 0 j n, it is easy to derive from Theorem 2.7 an asymptotic formula for the mixed Waring problem. The following result may be of independent interest.
where S(N) is given by (2.16).
There is relatively little in the literature concerning asymptotic formulae for R(N) for mixed exponents. The best result is due to Brüdern [3] who obtains an asymptotic formula for R(N) when m 0 = m 1 = 2, under some further conditions on the exponents, the most demanding of which is that Theorem 1.4 is not competitive with this, although it does not suffer from the defect that 2 must appear twice among the list of exponents. It remains an interesting open challenge to prove an asymptotic formula for R(N) for any value of n, when m i = 2 + i for 0 i n. When m = m 0 = · · · = m n , which is the traditional setting of Waring's problem, the condition in (1.2) reduces to n m 2 + m. This shows that our approach is not completely optimal in the equal exponent situation, since as explained in [17,Cor. 14.7], we know that n m 2 − m + O( √ m) variables suffice to get an asymptotic formula in Waring's problem. It seems likely that by combining methods developed by Wooley in [15] and [17, §14], one can recover this loss. (The authors are grateful to Professor Wooley for this remark.) Acknowledgements. While working on this paper the authors were both supported by EPSRC grant EP/P026710/1, and the second author received additional support from the NWO Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho Tanimoto for useful conversations related to this topic.

The Hardy-Littlewood circle method
We shall assume without loss of generality that 2 m 0 m 1 . . . m n . Our assumption (1.2) translates into In what follows it will be convenient to set Let N ∈ Z and let c = (c 0 , . . . , c n ) ∈ (Z\{0}) n+1 . Let H ∈ N, γ ∈ N n+1 and let h ∈ {0, 1, . . . , H − 1} n+1 . The main results in this paper are founded on an analysis of the counting function We shall view c as being fixed, once and for all, but γ can grow and so we will need all of our estimates to depend explicitly on it. In Theorems 1.1 and 1.3 we shall take N = 0 and c n = −1, whereas in Theorem 1.4 we take H = 1, for 0 j n. Then we may write Note that we may freely assume that γ j B for 0 j n, since otherwise M c;γ (B; h, H; N) = 0. Let δ be such that .
We define the major arcs M to be We define the minor arcs to be m = [0, 1)\M.

Contribution from the major arcs.
In the standard way we shall need to show that on the major arcs our exponential sums can be approximated by integrals, with acceptable error. The following result is a straightforward adaptation of familiar facts.
Proof. Let X ′ = (X − h)/H. If X ′ < q then the absolute value of the left hand side is trivially bounded by q + 1, and so we may proceed under the assumption that X ′ q. We write

The inner sum is
An application of the Euler-Maclaurin summation formula to this sum now yields the result.
. We apply Lemma 2.1 with X = B j , and α (resp. a) replaced by αc j γ j (resp. ac j γ j ). Thus αc j γ j − ac j γ j /q = βc j γ j and for any L > 1. Then it follows from Lemma 2.1 that Taking H 1 and observing that B j 1 for all 0 j n we see that On executing the sum over q we therefore conclude that It remains to analyse the terms S c;γ (B δ ; h, H; N) and J c (B δ ) .
Beginning with the singular series, it follows from [13, Theorem 7.1] that for any ε > 0. Therefore Let us define This is absolutely convergent, since (2.9) and (2.10) yield Moreover, Turning to the singular integral, it follows from [13, Lemma 2.8] that Thus, in view of (2.10), we deduce that is well-defined, and we have We are now ready to conclude our treatment of the major arcs. Note that On combining (2.6), (2.12) and (2.13), we therefore obtain the following result.

2.2.
Contribution from the minor arcs. According to work of Wooley [17, Eq. (1.8)], the main conjecture in Vinogradov's mean value theorem asserts that for each ε > 0 and t, k ∈ N, one has (2.14) This result was established recently by Bourgain, Demeter and Guth [1] using ℓ 2 -decoupling and also by Wooley [16,17] using efficient congruencing. The following mean value estimate is a straightforward consequence of their work.
Lemma 2.3. Let k ∈ N and let s be a real number satisfying s k(k + 1). Let A, H ∈ Z\{0} and h ∈ Z. Then we have where the implied constant does not depend on A, H or h.
Proof. Let 2t be the largest even integer such that 2t s. Then it follows that t k(k + 1)/2. By a trivial estimate and by considering the underlying equations of the following integrals via the orthogonality relation, we deduce where where α = (α k , . . . , α 1 ) and α ′ = (α k−1 , . . . , α 1 ). Summing trivially over n, the right hand side of our estimate is with the last equality an immediate consequence of considering the underlying equations of the integrals. An application of (2.14) now yields our result.
We also require the following Weyl type estimate, which is another consequence of the recent work on Vinogradov's mean value theorem. We omit the proof since it is obtained by invoking the main conjecture (2.14) in the proof of [14, Theorem 1.5].
Using this result we obtain the following bound for the exponential sum on the minor arcs. Proof. It will be convenient throughout the proof to write σ(m n ) = 1 m n (m n + 1) .
Let α ∈ m and let β = αc n γ n H mn . We put in this case. Thus we may suppose that B > 1.
By Dirichlet's theorem on Diophantine approximation we know there exist b ∈ Z and 1 r B such that gcd(b, r) = 1 and Note that b = 0 since α ∈ m. For simplicity let us write A = c n γ n H mn . We claim that bA > 0. But if bA < 0 then |β − b/r| = |αA − b/r| = α|A| + |b|/r > 1/r, since α > 0, which is a contradiction. This establishes the claim. Let A ′ = A/ gcd(A, b) and b ′ = b/ gcd (A, b).
Let X = (B n − h n )/H. First suppose B n /2H > X. Then B n < 2h n < 2H. In this case we clearly have S n (α) ≪ 1, which is satisfactory. Thus we suppose B n /2H X. In this case r B X mn and Lemma 2.4 yields for any ε > 0. Next, we note that If 2B 1−δ B/|c n γ n H mn 2 mn | it follows that On the other hand, if 2B 1−δ > B/|c n γ n H mn 2 mn | then We now verify that 1 |b ′ | r|A ′ |. We've already seen that |b ′ | 1, so we suppose that |b ′ | > r|A ′ |. Since α ∈ [0, 1) we have This is a contradiction, so that we do indeed have 1 |b ′ | r|A ′ |. We also have gcd(r|A ′ |, |b ′ |) = 1. Finally, α ∈ M if r|A ′ | B δ and B is sufficiently large, which is a contradiction. Therefore r|A ′ | > B δ and (2.15) becomes This completes the proof of the lemma, since σ(m n )−1/m n = −1/(m n +1).
We now have the tools in place to establish the following bound for the minor arc contribiution.
In the light of (2.1) we can assume that ℓ j m j (m j + 1) for all 0 j n − 1. It now follows from Hölder's inequality and Lemma 2.3 that since H 1 and γ j B for all 0 j n − 1. We apply Lemma 2.5 to estimate S n (α). The statement of the lemma follows on simplifying the final expression and observing that for all 0 j n − 1. We now turn to the task of proving an asymptotic formula for the counting function N(P n−1 , ∆; B) in Theorem 1.1. We shall assume without loss of generality that 2 m 0 . . . m n , so that (1.2) implies (2.1). The counting function can be written where we henceforth follow the convention that c n = −1. In view of (1.6), we may write Suppose that we are given vectors s and t with coordinates s j ∈ N and t j,r ∈ N for 0 j n and 1 r m j − 1. It will be convenient to introduce the set ∀ 0 j n µ 2 (v j,r ) = 1, gcd(v j,r , v j,r ′ ) = 1 c 0 x 0 + · · · + c n x n = 0 s j | u j and t j,r | v j,r ∀ j, r Given ǫ ∈ {±1} n+1 let ǫc denote the vector with coordinates ǫ j c j . Then where 1 is the vector with all coordinates equal to 1. We need to develop an inclusion-exclusion argument to cope with the coprimality condition in this expression. To ease notation we replace ǫc by c. Let x ∈ N c (B; 1, 1). It is clear that gcd(x 0 , . . . , x n ) > 1 if and only if there exists a prime p and a subset I ⊆ {0, . . . , n} for which p | u j for all j ∈ I and also p | m j −1 r=1 v j,r for all j ∈ I. (Note that I is allowed to be the empty set here.) Let G denote the set of all possible vectors g ∈ N n+1 with 1 g j m j − 1 for 0 j n. Let P = {2, 3, 5, . . .} denote the set of primes and let R be a non-empty finite collection of triples (g; p; I) where g ∈ G , p ∈ P and (possibly empty) I ⊆ {0, . . . , n}. Let R(p) be the subset of R containing all the triples in R with prime p. In what follows we adhere to common convention and stipulate that a union over the empty set is the empty set and a product over the empty set is 1. We let I(R(p)) = Next, we define a(R) to be the vector in N n+1 with coordinates and we define b(R) to be the vector in N n j=0 (m j −1) with coordinates b j,r = p∈P j∈J(g;R(p)) for some g∈G satisfying g j =r p, (0 j n, 1 r m j − 1).

(3.3)
It is easy to see that (a(R), b(R)) = (1, 1) as soon as R = ∅. Moreover, when R = {(g; p; I)} we see that N c (B; a(g; p; I), b(g; p; I)) is precisely the set of x ∈ N c (B; 1, 1) satisfying p | u j for all j ∈ I and p | v j,g j for all j ∈ I. In particular, it is now clear that  Proof. Let x belong to the intersection on the right hand side. Then, given any (g; p; I) ∈ R, we have p | u j for all j ∈ I and p | v j,r if j ∈ I and r = g j , where . Therefore, p | u j for all p such that j ∈ I(R(p)) and p | v j,r for all p such that j ∈ (g;p;I)∈R(p) g j =r {0, . . . , n}\I.
Thus (3.2) and (3.3) imply that a j | u j and b j,r | v j,r , whence it follows that x ∈ N c (B; a(R), b(R)). On the other hand, if x ∈ N c (B; a(R), b(R)) then we may reverse the argument to deduce that x also belongs to the intersection of all the sets N c (B; a(g; p; I), b(g; p; I)) for (g; p; I) ∈ R. This completes the proof of the lemma.
Given vectors s and t composed from positive integers, let Then, on combining the inclusion-exclusion principle with Lemma 3.1, we obtain It remains to asymptotically estimate these quantities. We collect together some properties of the function ̟(s, t).
Lemma 3.2. Let (s, t) = (1, 1) and let p ∈ P. We let s [p] be the vector whose jth coordinate is s j = p valp(s j ) and t [p] be the vector whose (j, r)th coordinate is t [p] j,r = p valp(t j,r ) . Then the following are true:  where we define multiplication of vectors by multiplying the corresponding coordinates. We clearly have (s, t) = p∈P (s [p] , t [p] ) and #R = p∈P #R(p). Thus ) .

It follows that
which thereby establishes (i).
To prove (ii) we note that it is not possible for p 2 to divide any coordinate of a(R(p)) or b(R(p)) for any prime p and R = ∅. Thus ̟(s [p] , t [p] ) = 0 if one of the coordinates of s [p] or t [p] is divisible by p 2 .
Finally, to prove (iv) we note there are only O(1) options for R(p) for any fixed p ∈ P. It now follows from the definition that Given (s, t) = (1, 1) and ε > 0, it follows from Lemma 3.2 that We begin by studying Let where the indices run over 0 j n and 1 r < r ′ m j − 1. For each s and t we let (1) v denote the sum over all v satisfying γ j B , gcd (v j,r t j,r , v j,r ′ t j,r ′ ) = 1 and µ 2 (v j,r t j,r ) = 1. (If there is no v which satisfies the above conditions then the sum is considered to be 0.) We may now write where M c;γ (B) = M c;γ (B; 0, 1; 0), in the notation (2.3). Guided by Lemma 3.2, we let (2) s,t denote the sum over (s, t) = (1, 1) satisfying s m j j m j −1 r=1 t m j +r j,r B and gcd (t j,r , t j,r ′ ) = 1, together with the condition that none of the coordinates of s or t is divisible by p 2 for any prime p and if one of the coordinates of s or t is divisible by p then p | s j t j,1 . . . t j,m j −1 for all 0 j n.
We want to apply Theorem 2.7 with H = 1 and N = 0. Let δ > 0 satisfy (2.5) and let S c;γ = S c;γ (0, 1; 0). Then, on appealing to Lemma 3.2 and (3.6), we deduce that (s,t) =(1,1) for any ε > 0, where Moreover, in view of (2.7) and (2.9), the error terms are given by We now need to estimate these three error terms. In doing so it will be convenient to set Using these estimates it follows that since n j=0 m j /(m j + 1) > 1. (Note that the factor 2m j − 1 on the right hand side comes from taking into account the 2m j − 1 possibilities where the factor p appears in s j or t j .) We have therefore shown that Turning to the estimation of F 2 (B), we may write We first show that if r 1. To see this we note that the left hand side is at most The inner x ′ -sum is absolutely convergent since r 1. The remaining sum over d | q is O(q ε ) for any ε > 0, by the standard estimate for the divisor function. This therefore establishes (3.9).
An application of (3.9) immediately yields for any ε > 0. Next, let  . In particular we have f 1 (q) f 1,1 (q). We claim that for any sufficiently small ε > 0. Once achieved, it will follow that To check the claim we let T denote the set of vectors (τ 0 , . . . , τ n ) ∈ N n+1 with the property that for any prime p we have val p (τ j ) ∈ {0, m j , . . . , 3m j − 1} and, furthermore, if p | τ 0 . . . τ n then val p (τ j ) > 0 for all 0 j n. Associated to any (τ 0 , . . . , τ n ) ∈ T is a unique choice for s, t. Thus we find that When p ∤ q the corresponding local factor takes the shape Alternatively, when p | q the factor is O(p 6ε(m 0 +···+mn) ) Assuming that ε is sufficiently small this therefore concludes the proof of (3.11). Finally we must analyse Applying (3.8) to handle the resulting sum over s and t it easily follows that We substitute our bounds for the error terms back into (3.7). This yields The error term is of the shape claimed in Theorem 1.1 and so it remains to analyse the quantity c B . The dependence on B in the factor c B arises from the definition of the sum (1) . A straightforward repetition of our arguments above suffice to show that for some η 2 > 0, where c is the constant that is defined as in c B , but with the summation conditions γ j B removed from (1) , for 0 j n. This shows that N(P n−1 , ∆; B) = cB Γ + O(B Γ−η ) for an appropriate η > 0, as claimed in Theorem 1.1. To go further, we adopt the notation s m w = (s m 0 0 w 0 , . . . , s mn n w n ), where we recall that w j = v Changing the order of summation, we may write with the understanding that ̟(1, 1) = 1 and t | v means t j,r | v j,r for all j and r. We claim that This will complete our analysis of the leading constant c in Theorem 1.1.
To check the claim we put c ′ j = ǫ j c j for 0 j n. It follows from (2.11) and multiplicativity that we deduce that Next, we put ̟(s, t) · #X p (s, t) .
On the other hand, on appealing to the inclusion-exclusion principle and the definition of ̟, for any prime p we return to (3.16) and see that ̟(s, t) · #X p,T (s, t).
Dividing by p nT and taking the limit T → ∞, we are now easily led to the proof of the claim (3.15).

Thin sets: proof of Theorem 1.3
Let Γ = n j=0 1 m j − 1, as in (2.2). In this section we assume that (1.2) holds and we let Ω ⊂ P n−1 (Q) be a thin set. Theorem 1.3 is concerned with an upper bound for the quantity gcd(x 0 , . . . , x n−1 ) = 1 |x| B, x i is m i -full ∀ 0 i n c 0 x 0 + · · · + c n−1 x n−1 = x n (x 0 : · · · : under the conditions on Ω that are stated in the theorem. Let us write N Ω (B) = N Ω (P n−1 , ∆; B) to ease notation. All of the implied constants in this section are allowed to depend on the thin set Ω. We shall proceed by using information about the size of thin sets modulo p on a set of primes p of positive density. Our thin set Ω is contained in a finite union t i=1 Ω i of thin subsets of type I and type II. We shall abuse notation and write Ω i (F p ) for the image of Ω i in P n−1 (F p ) under reduction modulo p. Similarly, we shall write Ω i (F p ) for the set of F p -points on the affine cone over this set of points.
Let Ω i ⊂ P n−1 (Q) be a thin subset of type I. Then it follows from the Lang-Weil estimates [8] that there exits C 1 > 0 such that for every sufficiently large prime p ∈ P Ω i , in the notation introduced before the statement of Theorem 1.3.
We take advantage of this information by noticing that gcd(x 0 , . . . , x n−1 ) = 1 |x| B, x i is m i -full ∀ 0 i n c 0 x 0 + · · · + c n−1 x n−1 = x n (x 0 : · · · : for any finite subset of primes S i . We stipulate that min p∈S i p is greater than some absolute constant depending only on n i=0 |c i |m i and the thin subset Ω i . Let The factor in front of (p − 1) is 1 when p ∈ Q m and at most m n * in general. The statement of the lemma now follows.
We are now ready to analyse the singular series in (4.7). Let us put c ′ j = ε j c j for indices 0 j n. We recall from (2.11) that  where N(p T ) = # k mod p T : In order to deal with primes p | H i , we require the following simple form of Hensel's lemma. Let p | H i . Then H i = pH ′ i for some H ′ i ∈ N that is coprime to p. It readily follows that N(p T ) = p n+1 # k mod p T −1 : n j=0 c ′ j w j (pk j + h j ) m j ≡ 0 mod p T .
If h mod p is a solution to the congruence c ′ 0 w 0 h m 0 0 + · · · + c ′ n w n h mn n ≡ 0 mod p, then necessarily it is a non-singular solution by (4.5), since each prime p | H i is large enough that p ∤ j c ′ j m j . Hence for T > 1 it follows from Lemma 4.3 that N(p T ) = p n+1 p n(T −1) = p nT +1 . Bringing this together with (4.8) and (4.9) for every prime p such that p | H i and p | w 0 . . . w n . Thus, on removing common factors of w j with H i , one easily concludes that . Once inserted into (4.10) and choosing S i in such a way that H i is a small enough power of B for (4.6), this shows that thin subsets of type I make a satisfactory overall contribution.
Suppose next that Ω i is type II. We may assume that p m * /κ for each p | H i . Then We choose S i to be set of primes 1 ≪ p log B/ log log B drawn from the set P Ω i ∩ Q m . In particular H i satisfies (4.6). Moreover, it follows from our assumption (1.4) that this set of primes has positive lower density ̺, say. But then Feeding this into the argument that we have just given yields from which it follows that the thin subsets of type II make a satisfactory overall contribution to (4.10) under the assumption (1.4). This completes the proof of Theorem 1.3.