Hyperbola method on toric varieties

We develop a very general version of the hyperbola method which extends the known method by Blomer and Br\"udern for products of projective spaces to complete smooth split toric varieties. We use it to count Campana points of bounded log-anticanonical height on complete smooth split toric $\mathbb{Q}$-varieties with torus invariant boundary. We apply the strong duality principle in linear programming to show the compatibility of our results with the conjectured asymptotic.


Introduction
This paper stems from an investigation of the universal torsor method [Sal98,FP16] in relation to the problem of counting Campana points of bounded height on log Fano varieties in the framework of [PSTVA21, Conjecture 1.1].Campana points are a notion of points that interpolate between rational points and integral points on certain log smooth pairs, or orbifolds, introduced and first studied by Campana [Cam04,Cam11,Cam15].The study of the distribution of Campana points over number fields was initiated only quite recently and the literature on this topic is still sparse [BVV12, VV12, BY21, Shu21, PSTVA21, Shu22, Str22, Xia22, BBK + 23].In this paper we deal with toric varieties, which constitute a fundamental family of examples for the study of the distribution of rational points [BT95b, BT95a, BT96, BT98, Sal98, dlB01a], via a combination of the universal torsor method with a very general version of the hyperbola method, which we develop.
We use the universal torsor method, instead of exploiting the toric group structure, because we hope to extend our approach to a larger class of log Fano varieties in the future.Indeed, the hyperbola method is well suited to deal with subvarieties [Sch14, Sch16, BB17, BB18, Mig15, Mig16, Mig18, BH19], and all log Fano varieties admit neat embeddings in toric varieties [ADHL15,GOST15] which can be exploited for the universal torsor method.
One of the key technical innovations in this article is the development of a very general form of the hyperbola method, which is motivated by work of Blomer and Brüdern in the case of products of projective spaces [BB18].Mignot [Mig18] has adjusted these ideas to complete smooth split toric varieties with simplicial effective cone.With our approach we extend the work of Blomer and Brüdern to complete smooth split toric varieties with additional flexibility to change the height function.
Let f : N s → R ≥0 be an arithmetic function for which one has asymptotics for summing the function f over boxes, see Property I in Section 4. Let B be a large real parameter, K a finite index set and α i,k ≥ 0 for 1 ≤ i ≤ s and k ∈ K.The goal is then to use this information from sums over boxes to deduce an asymptotic formula for sums of the form We define the polyhedron P ⊂ R s given by Here the parameters ̟ i , 1 ≤ i ≤ s are defined in Property I for the function f .The linear function s i=1 t i takes its maximal value on a face of P which we call F .We write a for its maximal value.
Theorem 1.1.Let f : N s → R ≥0 be a function that satisfies Property I from Section 4. Assume that P is bounded and non-degenerate, and that F is not contained in a coordinate hyperplane of R s .Let k = dim F .Then we have where C f,M and C f,E are the constants in Property I and c P is the constant in equation (4.3).
In comparison to earlier versions of the hyperbola method in work of Blomer and Brüdern [BB18] or Mignot [Mig18], we only obtain a saving of a power of a log B, but we can work with the weaker assumption of using only Property I.In contrast to [BB18] and [Mig18] we no longer need to assume that we can evaluate the function f on lower-dimensional boxes after fixing a number of variables, called Property II in their work.
The case treated in [BB18] would in our notation correspond to an index set K with one element where all the α i,k = α for all 1 ≤ i ≤ s and some α > 0, and k = s − 1.Our attack to evaluate the sum S f starts in a similar way as in [BB18].We cover the region given by the conditions s i=1 y α i,k i ≤ B, k ∈ K with boxes of different side lengths on which we can evaluate the function f .One important ingredient in the hyperbola method in [BB18] is a combinatorial identity for the generating series j1+...+js≤J ji≥0, 1≤i≤s t j1+...+js , which needs to be evaluated for J going to infinity.By induction the authors give a closed expression.For us this part of the argument breaks down, as we have in general more complicated polytopes that arise in the summation condition for S f and are not aware of comparable combinatorial identities for the tuples (j 1 , . . ., j s ) lying in general convex polytopes.Instead, we approximate the number of integer points in certain intersections of hyperplanes with a convex polytope by lattice point counting arguments and then use asymptotic evaluations for sums of the form 0≤m≤M m ℓ θ m for 0 < θ < 1 and ℓ, M ∈ N.
In comparison to Mignot's work [Mig18], we can deal with polytopes for which |K| > 1, and where k is no longer restricted to the case k = s − 1.
Our main application of Theorem 1.1 is a proof of [PSTVA21, Conjecture 1.1] for smooth split toric varieties over Q with the log-anticanonical height.
Theorem 1.2.Let Σ be the fan of a complete smooth split toric variety X over Q.Let {ρ 1 , . . ., ρ s } be the set of rays of Σ.For each i ∈ {1, . . ., s} fix a positive integer m i and denote by D i the torus invariant divisor corresponding to ρ i .Assume that L := s i=1 1 mi D i is ample.Let H L be the height defined by L as in Section 6.3.Let X be the toric scheme defined by Σ over Z, and for each i ∈ {1, . . ., s}, let D i be the closure of D i in X .For every B > 0, let N (B) be the number of Campana Zpoints on the Campana orbifold (X , where r is the rank of the Picard group of X, and c is a positive constant compatible with the prediction in [PSTVA21, §3.3]. Our application of the hyperbola method recovers Salberger's result [Sal98] and improves on the error term by saving a factor (log B) 1−ε where [Sal98] saves only a factor (log B) 1−1/f −ε , where f ≥ 2 is an integer that depends on the toric variety.
Theorem 1.2 could also be deduced from work of de la Bretèche [dlB01b], [dlB01a], who developed a multi-dimensional Dirichlet series approach to count rational points of bounded height on toric varieties, or from work of Santens [San23].Another approach could be via harmonic analysis of the height zeta function, even though such a proof would probably be more involved than the case of compactifications of vector groups [PSTVA21].Our proof proceeds via the universal torsor method introduced by Salberger in [Sal98] in combination with Theorem 1.1.One of our main motivations for this approach is that it opens a path to counting Campana points on subvarieties of toric varieties.
When we apply Theorem 1.1 to prove Theorem 1.2, we need to verify that both the exponent of B as well as the power of log B match the prediction in [PSTVA21].The exponent a in Theorem 1.1 is the result of a linear optimization problem.Similarly, the construction of the height function leading to the exponent one of B in Theorem 1.2 involves another linear optimization problem.We use the strong duality property in linear programming [Dan98,Chapter 6] to recognize that the exponents are indeed compatible, and that this holds heuristically also in the more general setting where the height is not necessarily log-anticanonical.For the compatibility of the exponents of log B we exploit a different duality setup, which involves the Picard group of X.
In upcoming work, we show how Theorem 1.1 can be used to count rational points and Campana points of bounded height on certain subvarieties of toric varieties.
The paper is organized as follows.In Section 2 we provide some auxiliary estimates on variants of geometric sums which are used later in Section 4. In Section 3 we study volumes of slices of polytopes under small deformations.In Section 4 we develop the hyperbola method and give a proof of Theorem 1.1.Sections 5 and 6 are dedicated to the application of the hyperbola method to prove Theorem 1.2.In Section 5 we study estimates for m-full numbers of bounded size subject to certain divisibility conditions, and we produce the estimates in boxes for the function f associated to the counting problem in Theorem 1.2.In Section 6 we describe the heights associated to semiample Q-divisors on toric varieties over number fields, we study some combinatorial properties of the polytopes that play a prominent role in the application of the hyperbola method, and we show that the heuristic expectations coming from the hyperbola method agree with the prediction in [PSTVA21, Conjecture 1.1] on split toric varieties for Campana points of bounded height, where the height does not need to be anticanonical.We conclude the section with the proof of Theorem 1.2.This article subsumes our previous work on the hyperbola method, which had appeared on arxiv.orgunder arXiv:2001.09815v2,and which proved Theorems 1.1 and 1.2 under additional technical assumptions on the corresponding polytopes.
1.1.Notation.We denote by N or Z >0 the set of positive integers.We use ♯S or |S| to indicate the cardinality of a finite set S. Bold letters denote s-tuples of real numbers, and for given x ∈ R s we denote by x 1 , . . ., x s ∈ R the elements such that x = (x 1 , . . ., x s ).For any subset S ⊂ R s we denote by cone(S) the cone generated by S.
We denote by F p the finite field with p elements and by F p an algebraic closure.For a number field K, we denote by O K the ring of integers, and by N(a) the norm of an ideal a of O K .We denote by Ω K the set of places of K, by Ω f the set of finite places, and by Ω ∞ the set of infinite places.For every place v of K, we denote by K v the completion of K at v, and we define | • | v = |N Kv /Qṽ (•)| ṽ , where ṽ is the place of Q below v and | • | ṽ is the usual real or p-adic absolute value on Q ṽ .We denote by | • | the usual absolute value on R.
We denote the Picard group and the effective cone of a smooth variety X by Pic(X) and Eff(X), respectively.For a divisor D on X we denote by [D] its class in Pic(X).We say that a Q-divisor D on X is semiample if there exists a positive integer t such that tD has integer coefficients and is base point free.

Preliminaries
In the hyperbola method in the next section we need good approximations for finite sums of the form for some 0 < θ < 1 and natural numbers ℓ, M ≥ 0 (here and in the following we understand 0 0 := 1).In this subsection we also write g ℓ (M ) for g ℓ (M, θ).
We will use the following result.
Lemma 2.1.For an integer ℓ ≥ 0 and 0 < θ < 1 and a real number M > ℓ we have Lemma 2.1 can be deduced from the following statement.
Lemma 2.2.Assume that M > ℓ ≥ 0 and θ > 0. Then we have For the proof of Lemma 2.2 and Lemma 2.1 we need the following identity.For an integer 0 ≤ α ≤ ℓ we have To see this consider the identity Now take derivatives with respect to t and then set t = 1.
Lemma 2.2 implies Lemma 2.1.We start in observing that We further compute Note that the term in the second line is equal to zero by equation (2.1).Hence we have We now switch the summation of m and h to obtain By the Faulhaber formulas 0≤t≤ℓ+1−h t ℓ is a polynomial in h with leading term Using equation (2.1) we hence obtain By equation (1.13) in [Gou72] we have We finish this section with a proof of Lemma 2.2.
Proof of Lemma 2.2.We compute We split the last summation into three ranges depending on the size of m and get We now use the identity (2.1) for the third last line and deduce that Now the lemma follows in expanding each of the terms (M + m − h) ℓ .

Volumes of certain sections of polytopes
In this section we provide some estimates on the volumes of intersections of convex polytopes with certain hyperplanes.These will be used in the next section in the development of our generalized form of the hyperbola method.The vector space R s is endowed with a fixed inner product that is used to define orthogonality and the Lebesgue measures.Proposition 3.1.Let P ⊆ R s be an s-dimensional convex polytope with s ≥ 1.Let F be a proper face of P. Let H ⊆ R s be a hyperplane such that H ∩ P = F .Let w ∈ R s such that P ⊆ H + R ≥0 w.For δ > 0, let H δ := H + δw.Let k := dim F .We denote by meas j the j-dimensional measure induced by the Lebesgue measure on R s .Then (i) for δ > 0 sufficiently small, where c is a positive constant that depends on P, F and H. (ii) If s ≥ 2, for δ > 0 sufficiently small we have where c ′ is a positive constant that depends on P, F and H. Let T ⊆ R s be a hyperplane such that T ∩ P is a face of P. Let d := dim P ∩ T .Let u ∈ R s be a vector such that P ⊆ T + R ≥0 u.For κ > 0, let T κ = T + κu and T ≤κ = T + [0, κ]u.Then (iii) for δ and κ sufficiently small and positive, where the implicit constant is independent of κ and δ. (iv) For δ and κ sufficiently small and positive, where the implicit constant is independent of κ and δ. (v) If s ≥ 2, for δ and κ sufficiently small and positive, where the implicit constant is independent of κ and δ.
Let V ⊆ R s be an affine space of dimension m with k ≤ m ≤ s − 1.We denote by p V : R n → V the orthogonal projection onto V .Then (vi) for δ and κ sufficiently small and positive such that δ ≤ κ, where the implicit constant is independent of κ and δ.
Hence we can assume that F ∩ T = ∅.
Step 1.We first prove the case where P is a simplex.We denote by v 0 , . . ., v k the vertices of F and by v k+1 , . . ., v s the vertices of P not contained in F .Since F ∩ T = ∅, we can assume that v 0 ∈ T .Up to a translation, which is a volume preserving automorphism of R s , we can assume that v 0 is the origin of R s .We observe that v 1 , . . ., v s form a basis of the vector space R s .Let C be the cone (with vertex v 0 ) generated by v k+1 , . . ., v s .We observe that C ∩ H = {v 0 } as H ∩ P = F and C is contained in the cone (with vertex v 0 ) generated by P.
We prove (i) by induction on k.If k = 0, for δ small enough, and hence s ≥ 2. Let L ⊆ R s be the hyperplane that contains v 1 , . . ., v s .Then L + = L + R ≤0 v 1 is the halfspace with boundary L that contains v 0 , and w be the half space with boundary H 1 that contains F .Then H + 1 ∩ Q is bounded, and for δ small enough, an s-dimensional polytope that intersects H in the (k − 1)dimensional face with vertices v 1 , . . ., v k .By induction hypothesis we have as F ⊆ H. Hence, there is a positive constant a (which is the determinant of the matrix of a suitable linear change of variables in H) such that We conclude the proof of (i) for P simplex as For part (iii), we observe that where the implicit constant is independent of κ and δ.Since P ⊆ T + R ≥0 u and P T , there is j ∈ {1, . . ., s} such that (R ≥0 v j ) ∩ T ≤κ is bounded, i.e., (R ≥0 v j ) ∩ T ≤κ = [0, 1]κa j v j for some a j > 0 independent of κ.Let T ≤κ,j := s i=1 λ i v i : 0 ≤ λ j ≤ κa j .
If F ⊆ T , we can choose j ≤ k.Then F ∩ T ≤κ ⊆ F ∩ T ≤κ,j .Since F is a simplex (it is a face of a simplex), where F j is the maximal face of F that does not contain v j .Moreover, where the implicit constant is independent of κ and δ.
For part (iv), we observe that If F ⊆ T , and d > k.Let C ′ = C∩T .Up to rearranging the indices {k+1, . . ., s}, we can assume that C ′ is the cone (with vertex v 0 ) generated by v k+1 , . . ., v d .Let C ′′ be the cone (with vertex v 0 ) generated by v d+1 , . . ., v s .Then C = C ′ + C ′′ , and , where p is the projection onto H 1 along v d , as If F ⊆ T , let F ′ = F ∩ T .Up to rearranging the indices {1, . . ., k} we can assume that the vertices of the simplex F ′ are v 0 , . . ., v a , where a = dim F ′ .Let C ′ be the cone (with vertex v 0 ) generated by v a+1 , . . ., v k .Then F ⊆ F ′ + C ′ ⊆ H, and hence Then where p is the projection onto T 1 along v a+1 , as Now we turn to part (vi).Up to translating V along an orthogonal direction, we can assume that v 0 ∈ V .We denote by V ⊥ the affine space through v 0 orthogonal to V .If p V | H δ is not surjective, then dim p V (H δ ∩ P ∩ T ≤κ ) < m and hence meas m p V (H δ ∩ P ∩ T ≤κ ) = 0. Thus we can assume without loss of generality that

If F
V ⊥ , there exists j ∈ {1, . . ., k} such that v j / ∈ V ⊥ .Without loss of generality we can assume that j = 1.If F T there exists If C V , we can assume without loss of generality that e s / ∈ V .Then there is Let e m+1 , . . ., e s be a basis of V ⊥ with e s = e.Then there is a set . ., e s } is a set of linearly independent vectors, and {v i : i ∈ I} ∪ {v, e m+1 , . . ., e s } is a set of linearly independent vectors if F V ⊥ .Without loss of generality we can assume that I ⊇ {k + 1, . . ., m}.Let e i = p V (v i ) for i ∈ {k + 1, . . ., m} and complete to a basis e 1 , . . ., e m of V with Recalling the definition of v, we conclude that if C V Thus, recalling the definition of v, we conclude that for δ < 1 the bounds (3.2) hold also if C ⊆ V .Thus we proved (i), (iii), (iv), and (vi) for P simplex.
Step 2. We complete the proof of (i), (iii), and (iv).If P is not a simplex, let P 1 , . . ., P N ⊆ R s be simplices of dimension s such that P = N i=1 P i is a triangulation of P. Then Therefore, parts (i), (iii) and (iv) follow from Step 1.If δ ≤ κ, (vi) follows from Step 1.
Step 3. It remains to prove (ii) and (v).Let F 1 , . . ., F M be the faces of P. Then If k = 0, then F is a vertex of P. Up to a translation, which is a volume preserving automorphism of R s , we can assume that F is the origin of R s .We denote by C the cone with vertex F generated by P.Then, for δ small enough, Moreover, for δ small enough we can assume that H δ does not contain any vertex of P. Since , where the sum actually runs over the maximal faces of P that intersect F .Let F be an (s − 1)-dimensional face of P that intersects F such that F = F , and let k = dim( F ∩ F ).By part (i) applied replacing P by F , we have meas For δ and κ small enough, we have Moreover, for δ and κ small enough we can assume that H δ does not contain any vertex of P and T ≤κ does not contain any vertex of P not in P ∩ T .Since meas s−2 (H δ ∩ F i ∩ T ≤κ ) = 0 whenever H δ ∩ F i ∩ T ≤κ has dimension strictly smaller than s − 2, we have where the sum actually runs over the maximal faces of P that intersect F ∩ T .
Let F be an (s − 1)-dimensional face of P that intersects F ∩ T such that F = F , and let k = dim( F ∩ F ).By part (iii) and (i) applied replacing P by F , we have Then part (v) follows by part (iv).
In the next section we apply the proposition above to the polytope P in Theorem 1.1 with ≤ κ} for given i 0 ∈ {1, . . ., s}, and P i0,κ = P ∩T ≤κ .Since P is a full dimensional polytope, for δ sufficiently small, where c P is a positive constant depending only on P and H. Since the face F = H ∩ P is not contained in any coordinate hyperplane of R s , we have F T and hence and for V any coordinate subspace of dimension m ≥ k, Notice that if m = k, then the volume of the projection is independent of the choice of δ, hence holds also for δ = 0.

Hyperbola method
We consider a function f : N s → R ≥0 with the following property.
Property I: Assume that there are non-negative real constants C f,M ≤ C f,E and ∆ > 0 and where the implied constant is independent of f .
Let B be a large real parameter, K a finite index set and s ∈ N. Let α i,k ≥ 0 for 1 ≤ i ≤ s and k ∈ K.
Our goal is to evaluate the sum (if finite) We start with a heuristic for the expected growth of the sum S f (B).Let B be a large real parameter.Consider the contribution to the sum S f (B) from a dyadic box where each y i ∼ B ti̟ −1 i say for real parameters t i ≥ 0 (for example we could think of to the sum S f (B).In order for such a box to lie in the summation range we roughly speaking need and The system of equations (4.1) and (4.2) defines a polyhedron P ⊂ R s .We make the following two assumptions on P.
Assumption 4.1.We assume that P is bounded and non-degenerate in the sense that it is not contained in a s − 1 dimensional subspace of R s .
Assumption 4.2.The face F on which the function s i=1 t i takes its maximum on P is not contained in a coordinate hyperplane of R s (with coordinates t i ).

The linear function
takes its maximum on a face of P. We call the maximal value a and assume that this maximum is obtained on a k-dimensional face of P.
Let H δ be the hypersurface given by It comes equipped with an s − 1 dimensional measure which is obtained from the pull-back of the standard Lebesgue measure to any of its coordinate plane projections.In the following we write meas for this measure.Note that Proposition 3.1(i) shows that Assumption 4.1 implies that the following holds.
There is a constant c P such that for δ > 0 sufficiently small (in terms of P) we have for a sufficiently large constant C, depending only on P.
Let l ∈ Z s ≥0 and 1 < θ ≤ 2 a parameter to be chosen later.We define box counting functions Assume that f satisfies Property I. Then by inclusion-exclusion we evaluate Recall that we assumed in Property I that We note that the sum Let Ã be a large natural number, which we view as a parameter to be specified later.
We set S 1,f := We now cover the sum S 1,f with boxes of the form B f (l, θ).Let L + be the set of l ∈ Z s ≥0 such that the following inequalities hold Similarly, let L − be the set of l ∈ Z s ≥0 such that the following inequalities hold Let C 5 be a positive constant such that We define L − to be the set of l ∈ Z s ≥0 such that the following inequalities hold Then we have where we read the last line as a definition for S − 1,f and S + 1,f .Note that the coverings into boxes do not depend on the function f but only on the summation conditions on the variables y i , 1 ≤ i ≤ s.
Let r + (l) (resp.r − (l)) be the set of l ∈ L + (resp.L − ) such that We recall that This leads to Every vector l ∈ L − satisfies the bound and hence This leads to the bound We deduce that Let r(l) be the number of l ∈ Z s ≥0 such that s i=1 l i = l and the following inequalities hold Note that r(l) is the number of lattice points in the polytope given by and Note that we have We now stop a moment to introduce some more auxiliary polytopes.We recall that P ⊂ R s is the polytope given by the system of equations (4.1) and (4.2).
For 1 ≤ i 0 ≤ s and κ > 0 we introduce the polytope P i0,κ given by the system of equations and t i0 ≤ κ.
I.e.P i0,κ is obtained from intersecting P with the halfspace t i0 ≤ κ.Let H δ be defined as before, i.e. the hyperplane given by By Proposition 3.1(iii) and under the Assumption of 4.1 and 4.2 we have the following property.
Let 0 ≤ k ≤ s − 1. Assume that κ > 0 and δ > 0 are sufficiently small in terms of the data describing P. Then we have meas(H δ ∩ P i0,κ ) ≪ κδ s−1−k . (4.5) Remark 4.3.Note that in the case k = 0 and where the maximal face F is not contained in a coordinate hyperplane, the intersection H δ ∩ P i0,κ is empty for ǫ and δ sufficiently small.
In our applications it will take κ of size κ ≪ log log B log B .We next evaluate the function r − (l) asymptotically.
Lemma 4.4.Assume that 0 ≤ k ≤ s − 1. Assume that Assumption 4.1 and Assumption 4.2 hold.Let l be an integer with for δ sufficiently small, depending only on P, as in Proposition 3.1(i),(iii).
Then we have Here we read 0 then we have r − (l) = 0.
Remark 4.5.Note that exactly the same asymptotic also holds for r + (l), but then in the range Proof.We recall that r(l) counts lattice points in the polytope P (l, B, θ) given by s i=1 We observe that P (l, B, θ) is equal to the polytope log B log θ − C 5 P, i.e. the polytope P blown up by a factor of log B log θ −C 5 , intersected with the hyperplane log B log θ − C 5 H δ ′ given by where δ ′ ≥ 0 is chosen such that I.e.our task is to count lattice points in the polytope log B log θ − C 5 (P ∩ H δ ′ ), which is the same as counting integer lattice points in the projection of this polytope to one of the coordinate hyperplanes.Here we can apply Davenport's lemma [Dav51].
By equation (4.3) we have We can rewrite this as The measure of the projection of P ∩H δ ′ to various coordinate spaces is bounded.Hence the measure of the projections of dimension at most s − 2 of the blown-up polytope log B log θ − C 5 (P ∩ H δ ′ ) is bounded by . By Davenport's lemma [Dav51] and for δ sufficiently small we find that Finally, by the same arguments as before and equation (4.5) we have for any 1 Here we use again that all volumes of projections of the corresponding polytope are bounded.
Let δ 0 > 0 be a parameter.We write First we bound the error term E − 3,f .For this we observe that if l ≤ (a−δ 0 ) log B/ log θ, then Moreover, as each of the l i in the counting function r − (l) is bounded by ≪ log B log θ we have This gives the estimate We conclude that as long as We now use Lemma 4.4 to first evaluate the main term M − 1,f .For B sufficiently large and δ 0 sufficiently small, we have We can also write M − 1,f as In the second line we computed the geometric series.In the following we assume that for B large we choose θ in a way such that a log B log θ is an integer.Under this assumption M − 1,f becomes We further rewrite this as We recall the notation We now apply Lemma 2.1 and obtain Next we need to choose θ.We assume that where A > s is a fixed parameter.Then we have Moreover we assume that we now take δ 0 sufficiently small such that Lemma 4.4 holds.Note that for B sufficiently large, we automatically have 20s(A + 1) log log B ≤ δ 0 log B.
We deduce that We now turn to the treatment of the error term.Recall that we have We now observe under the assumption of equation (4.7) that we have Let A † be a positive real parameter.If we take Ã sufficiently large depending on A, A † , s and ∆, then we get Observe that the same calculations are also valid for S + 1,f in place of S − 1,f .We deduce that We recall that we have made the assumption that a log B/ log θ is integral.Hence we need to show that for every B sufficiently large there is a θ in the range (4.7) such that this expression is integral.Note that the conditions on θ in (4.7) translate into saying that which for B growing certainly contains an integer.More generally, let W i > 0, 1 ≤ i ≤ s be parameters such that there exists N > 0 with We rephrase our findings in the following lemma.
Lemma 4.6.Let f : N s → R ≥0 be a function satisfying Property I. Assume that Assumption 4.1 and Assumption 4.2 hold.Then, for any Ã sufficiently large in terms of s and ∆, we have where the implied constants may depend on N , Ã and the polytope P.
Next we turn to the treatment of the contributions where some variables in the sum S f (B) can be small.Let N > 0 be a parameter as above and consider W < (log B) N .For i 0 ∈ {1, . . ., s} set We cover the region of summation by dyadic boxes, i.e. we set θ = 2 above.Note that for any value of l we have We define the set L i0 to be the set of lattice points l ∈ Z s ≥0 which lie in the polytope given by and Let r W,i0 (l) be the number of l ∈ Z s ≥0 in L i0 with s i=1 l i = l.With this notation we find that We observe that r W,i0 (l) is the number of lattice points in the intersection of the polytope log B log θ P i0,κ , where κ = ̟i 0 log W log B , intersected with the hyperplane log B log θ H δ with δ = a − log θ log B l. Let δ 1 > 0 be a parameter to be chosen later.As above, we observe that Hence, if we assume that (s + 1) log log B ≤ δ 1 log B, (4.8) then we have If k = 0, then for κ and δ sufficiently small the intersection of H δ and P i0,κ is empty and there is nothing to bound.Hence in the following we may assume k ≥ 1.
By enlarging B by at most a constant factor depending on a, we may assume that a log B log θ is integral.If l = log B log θ a, then δ = 0 and the dimension of the intersection of P i0,κ ∩ H 0 is at most k.If the dimension of the intersection of P i0,κ ∩ H 0 is at most k − 1, then we observe that all projections of this intersection to coordinate spaces are bounded, and hence we obtain the bound Finally, assume that the dimension of the intersection P i0,κ ∩ H 0 is equal to k.Consider the projection of P i0,κ ∩ H 0 to a k-dimensional coordinate subspace V .Then we aim to bound the number of integer lattice points which are contained in the projection of log B log θ (P i0,κ ∩ H 0 ) to V .By Proposition 3.1(vi) the k-dimensional volume of the projection of P i0,κ ∩ H 0 to V is bounded by ≪ κ and the lower dimensional volumes of projections to coordinate spaces of dimension at most k − 1 are bounded.
Then, by Davenport's lemma [Dav51] we have Next we consider the case log B log θ (a − δ 1 ) ≤ l < log B log θ a, i.e. 0 < δ ≤ δ 1 .As we are only interested in an upper bound for S i0 (W, B), we may enlarge W to W = (log B) N with N sufficiently large, such that we have Then we can choose δ 1 such that (4.8) is satisfied and such that δ 1 ≤ κ.By Proposition 3.1(vi) the polytope P i0,κ ∩ H δ has the following properties if κ is sufficiently small and δ ≤ κ: (i) If we project P i0,κ ∩ H δ to coordinate spaces of dimension k − 1 or smaller, then the volume is bounded by an absolute constant.(ii) Let k − 1 < m ≤ s − 1.Then the m-dimensional volume of the projection of P i0,κ ∩ H δ to any m-dimensional coordinate space is bounded by Then by Davenport's lemma [Dav51] we obtain Again using that θ = 2 we obtain.
By Lemma 2.1 or in observing that the last sum is absolutely convergent, we find that This completes the proof of Theorem 1.1.

m-full numbers
Let m ≥ 1 be a natural number.We recall that an integer y is called m-full if for each prime divisor p of y, we have that p m divides y.We introduce the function that counts the number of m-full natural numbers less than BG58]).For each m ≥ 1 and B > 0 we have (5.1) where C 1 = 1, κ 1 = 0, and for m ≥ 2, For a square-free positive integer d, we define (5.3) In this section we prove an asymptotic formula for the function F m (B, d).We will first do it for the case that d is a prime.We will then inductively on the number of prime factors of d provide a general asymptotic formula.First we provide a form of inclusion-exclusion lemma, which expresses F m (B, p) for a prime number p in terms of sums of the function F m (B).Before we state the lemma, we introduce a convenient piece of notation.For r ≥ 1 and k ∈ Z let Lemma 5.2.For m ≥ 2 one has Note that the summations in Lemma 5.2 are in fact finite, as F m (P ) = 0 if P < 1.
Proof.We start the proof in reinterpreting terms of the shape F m Bp −K for some K > 0 as Then the right hand side in the identity in Lemma 5.2 becomes (5.4)For any K ≥ 0 we use the identity We use this identity for the terms in the third line in (5.4).We observe that the terms counting 1 ≤ p 2rm+k l ≤ B with l m-full identically cancel with the fourth line in (5.4).Hence we obtain Recalling the definition of the functions ρ m (k, r) we can further rewrite this as (5.6) We now use the identity (5.5) for the terms in the last line in equation (5.6).The resulting terms with p 2rm+k l ≤ B and p ∤ l cancel with the terms in the fourth line.Moreover, the terms with 1 ≤ p (2r+1)m+k l ≤ B and l m-full identically cancel with the terms in the second line in (5.6).Hence we obtain Again using the definition of the functions ρ m (k, 2r) we can rewrite this as (5.7) The last two sums in (5.7) cancel except for the terms with r = 1 in the second line.Hence we get Similarly as in (5.5) we now observe that on the right hand side we count exactly all 1 ≤ l ≤ B such that p | l and l is m-full, which completes the proof of the lemma.
Lemma 5.2 now allows us to deduce an asymptotic formula for F m (B, p) given that we know (5.1).We recall that the sums in Lemma 5.2 are all finite and hence we can first reorder them to take into account cancellation between different sums and then complete the resulting series to infinity.First we rewrite the expression for F m (B, p) in Lemma 5.2 as (5.8) with coefficients a m (µ) that are given by Note that all the appearing sums are in fact finite and hence we can reorder them freely.Our next goal is to get more understanding on the coefficients a m (µ) (in particular their size) and hence we group them in a generating series.Define A first rough bound on the coefficients a m (µ) can be obtained via the estimate For m ≥ 3 we obtain a m (µ) ≤ 2(m − 1) µ m +1 , whereas for m = 2 we have the estimate In particular we deduce that there is some constant R m only depending on m such that the power series G m (x) is absolutely convergent for |x| < R m .Moreover, in choosing R m sufficiently small we can also assume that the sum is absolutely convergent.Our next goal is to write the generating series G m (x) as a fractional function and in this way realise that it has a larger radius of absolute convergence than the bound that is obtained from the very rough estimate on a m (µ).For this we observe that for |x| < R m we can express G m (x) as We can now compute the generating function G m (x) as In the area of absolute convergence one may reorder the sums as We observe that Hence we obtain In the interval x ∈ (0, 1) the function x m − x takes its minimum at x = m − 1 m−1 , and at this point . In particular we observe that the Taylor series for G m (x) is absolutely convergent in the interval x ∈ (0, 1).
We can now deduce an asymptotic for F m (B, p).Lemma 5.3.Let m ≥ 2. Let p be a prime number.Then we have Here the implicit constant is independent of p.
Proof.We start in recalling equation (5.8) From the asymptotic formula in (5.1) we deduce that The last sum is absolutely convergent (consider the generating function x −m G m (x) at the point x = 2 −κm ) and hence we have established the asymptotic Next we aim to generalize Lemma 5.3 to obtain an asymptotic formula for F m (B, d) for a general square-free number d.For this we start with a generalization of Lemma 5.2.
The proof of Lemma 5.4 is exactly the same as the proof of Lemma 5.2 where the condition l is m-full is replaced by the condition that l is m-full and d ′ | l.Moreover, as in equation (5.8) one can rewrite the identity from Lemma 5.4 as (5.9) Via induction on the number of prime factors of d we now establish the following lemma.
Lemma 5.5.Let d > 0 be a square-free integer.Write ω(d) for the number of prime divisors of d.Then for each integer m ≥ 2 there exists a positive constant K m such that we have Here the implicit constant is independent of d and (5.10) For m = 1 the asymptotic holds with Proof.For m = 1 the statement is immediate.Let us assume that m ≥ 2. If d is prime, then the statement follows from Lemma 5.3 (or note that if d = 1 then the statement reduces to the assumption in (5.1)).Let d > 0 be squarefree and q a prime with q | d.Assume that we have established the asymptotic with a constant K m given by Note that K m is indeed a convergent sum.Then by Lemma 5.4 and equation (5.9) we deduce that By definition of K m we obtain Next we given an upper bounds for the leading constant in Lemma 5.5.For squarefree d we introduce the notation (5.11) We observe that c 1,d = 1/d and we recall that (5.12) For every fixed m ≥ 1 there is a positive constant c 2 (m) < 1 such that holds for all primes p ≥ 2. Hence we deduce from equation (5.12) that there exists a constant c 3 (m), only depending on m, such that (5.13) Hence, we get (5.14) Moreover, for m ≥ 2 we have Proof.We observe that 1≤yi≤Bi,1≤i≤s For i ∈ {1, . . ., s} such that m i ≥ 2 apply Lemma 5.5, for i ∈ {1, . . ., s} such that ).Then apply (5.14) and (5.15) to estimate the error term.
Lemma 5.6 implies that the function f m,d satisfies Property I with the constants , and 1/(m i (m i + 1))}.
6. Campana points on toric varieties 6.1.Toric varieties over number fields.Let X be a complete smooth split toric variety over a number field K. Let T ⊆ X be the dense torus.Let Σ be the fan that defines X.We denote by {ρ 1 , . . ., ρ s } the set of rays of Σ and by Σ max the set of maximal cones of Σ.For every maximal cone σ we define J σ to be the set of indices i ∈ {1, . . ., s} such that the ray ρ i belongs to the cone σ, and we set I σ = {1, . . ., s} J σ .Then we have |J σ | = n and |I σ | = r for every maximal cone σ of Σ, where n is the dimension of X and r is the rank of the Picard group of X.In particular, s = n + r.For each i ∈ {1, . . ., s}, we denote by D i the prime toric invariant divisor corresponding to the ray ρ i .We fix a canonical divisor K X := − s i=1 D i .By [Cox95] the Cox ring of X is K[y 1 , . . ., y s ] where the degree of the variable y i is the class of the divisor D i in Pic(X).For every y = (y 1 , . . ., y s ) ∈ C s and every D = s i=1 a i D i , let Let Y → X be the universal torsor of X as in [Sal98,§8].We recall that the variety Y is an open subset of A s K whose complement is defined by y Dσ = 0 for all maximal cones σ, where D σ := i∈Iσ D i for all σ ∈ Σ max .
The integral model π : Y → X of the universal torsor Y → X as in [Sal98, Remarks 8.6] gives a parameterization of the rational points on X via integral points in O s K = A s (O K ) as follows.Let C be a set of ideals of O K that form a system of representatives for the class group of K. We fix a basis of Pic(X), and for every divisor D on X we write c D := (6.1) Let N be the lattice of cocharacters of X.Then Σ ⊆ N ⊗ Z R. For every i ∈ {1, . . ., s}, let ν i be the unique generator of ρ i ∩ N .For every torus invariant divisor D = s i=1 a i D i of X and for every σ ∈ Σ max , let u σ,D be the character of N determined by u σ,D (ν j ) = a j for all j ∈ J σ , and define D(σ) := D − s i=1 u σ,D (ν i )D i .Then D and D(σ) are linearly equivalent.For every i, j ∈ {1, . . ., s}, let β σ,i,j := −u σ,Dj (ν i ).Then, for every i, j ∈ {1, . . ., s}, we have β σ,i,j = 0 whenever j ∈ I σ , and whenever i = j are both in J σ .Hence, (6.2) Lemma 6.1.For every i, j ∈ {i, . . ., s} and σ, σ ′ ∈ Σ max we have Remark 6.5.If L is ample, then there exists a positive integer t such that [t −1 L] = s i=1 1 mi D i with m 1 , . . ., m s ∈ Z >0 .Indeed, [L] has at least one representative of the form s i=1 b i D i with b 1 , . . ., b s ∈ Q >0 by Lemma 6.4 and Remark 6.2.Hence, it suffices to choose any positive integer t such that tb −1 i ∈ Z for all i ∈ {1, . . ., s}. 6.2.2.The polytope P .We investigate some polytopes associated to [L] that we use for the application of the hyperbola method in Sections 6.5-6.6.
We identify R s with the space of linear functions on s i=1 RD i by defining t(D i ) = t i for all i ∈ {1, . . ., s} and all t = (t 1 , . . ., t s ) ∈ R s .Under this identification, the dual of Pic(X) is the linear subspace H of R s defined by for one, or equivalently all, σ ∈ Σ max (cf.Lemma 6.1).
Let P ⊆ R s be the polyhedron defined by t i ≥ 0 ∀i ∈ {1, . . ., s} and Then P is a full dimensional convex polytope by Remark 6.2 and Assumption 6.3.Moreover, cone( P ) is dual to the cone C defined above.For every σ ∈ Σ max , let We observe that the polytopes P and P σ depend only on the class of L in Pic(X) and not on the chosen representative s i=1 a i D i .
In particular, F ∩ {t 1 , . . ., t s > 0} = ∅.Proof.Since s ≥ 1 and P is full dimensional, we have a(L, ̟) > 0. For every σ ∈ Σ max , let F σ := F ∩ P σ .Fix σ ∈ Σ max .Let t ∈ P σ H.By (6.4) and Lemma 6.6 there exists j ∈ J σ such that t j < i∈Iσ β σ,i,j t i .Let t ′ j := i∈Iσ β σ,i,j t i .For each i ∈ {1, . . ., s} {j} let t ′ i := t i .Then (t ′ 1 , . . ., t ′ s ) ∈ P σ , and For (ii) we recall that α i,σ = ̟ i + j∈Jσ ̟ j β σ,i,j for all i ∈ I σ .Hence, we have i∈Iσ α i,σ t i = s i=1 ̟ i t i for all t ∈ H and for all σ ∈ Σ max .Since H is the subspace of R s dual to Pic(X) R , a torus invariant divisor D satisfies t(D) = 0 for all t ∈ H if and only if D is a principal divisor.Since D 1 , . . ., D s are not principal divisors, then H ∩ {t 1 , . . ., t s > 0} = ∅.Let t ∈ H with t 1 , . . ., t s > 0, up to rescaling t by a positive real number we can assume that s i=1 ̟ i t i = 1, and hence t ∈ F .6.2.3.The geometric constant.We compute certain volumes of polytopes that appear in the leading constant of the asymptotic formula 1.1.
Fix σ ∈ Σ max .Since X is smooth, we know that Pic(X) = i∈Iσ Z[D i ].We identify R r with the space of linear functions on i∈Iσ R[D i ] by defining for all integrable functions g : R r → R. Lemma 6.8.The volume is positive and independent of the choice of ĩ and of the choice of σ.

6.3.
Heights.Now we study the height associated to a semiample Q-divisor L on X.Let t be a positive integer such that tL has integer coefficients and is base point free.By [CLS11, Proposition 4.3.3]we have H 0 (X, tL) = m∈PtL∩M Kχ m , where P tL and M are defined in Section 6.2 and χ m ∈ K[T ] is the character of T corresponding to m.Let H tL : X(K) → R ≥0 be the pullback of the exponential Weil height under the morphism X → P(H 0 (X, tL)) defined by the basis of H 0 (X, tL) corresponding to P tL ∩ M .We define H L := (H tL ) 1/t .We observe that this definition agrees with [BT95a, §2.1].Proposition 6.10.For every y ∈ Y (K), we have Let m ∈ P tL ∩M .By (6.3) there are λ σ ∈ R ≥0 for σ ∈ Σ max such that σ∈Σmax λ σ = 1 and m = − σ∈Σmax λ σ u σ,tL , so that y tL+(χ m ) = y σ∈Σmax λσ tL(σ) .This proves the first statement.For the second statement we argue as in the proof of [Pie16, Proposition 2].Fix c ∈ C r and y ∈ Y c (O K ).For every prime ideal p of O K we write v p for the associated valuation.Then where the first equality holds as [L(σ)] = [L] in Pic(X) R , and the second equality follows from (6.1) as y L(σ) ∈ c L(σ) for all σ ∈ Σ max .
The following lemma will ensure the Northcott property for H L .Lemma 6.11.If L satisfies Assumption 6.3, then there is α > 0 such that for every c ∈ C r and B > 0, every point y ∈ (ii) Assume, additionally, that L is ample.Fix σ ∈ Σ max and define γ i := j∈Jσ βσ,i,j mj .Then a(L) is the minimal value of the function λ 0 subject to the conditions λ 0 , λ j ≥ 0, ∀j ∈ J σ , (6.12) Proof.To prove part (i) we observe that for t ∈ R the condition and C is a cone.Then (6.14) holds if and only if there exists a divisor D ′ ∈ (tL + M R ) ∩ C such that D ′ + K X + ∆ ∈ C. By Lemma 6.4 this is equivalent to the existence of λ σ ∈ R ≥0 for all σ ∈ Σ max such that σ∈Σmax λ σ = t and σ∈Σmax λ σ L(σ) + K X + ∆ ∈ C. Now we prove part (ii).Condition (6.14) is equivalent to the existence of D ∈ C such that t[L] + [K X + ∆] = ϕ(D) in Pic(X) R .Since X is proper and smooth, the last equality is equivalent to tL(σ) + (K X + ∆)(σ) = D(σ).Write D = s i=1 λ i D i .Then D ∈ C if and only if λ 1 , . . ., λ s ≥ 0. We have Using the fact that λ i ≥ 0 for all i ∈ I σ if D ∈ C, we see that condition (6.14) is equivalent to the existence of λ j ∈ R ≥0 for all j ∈ J σ that satisfy the conditions in the statement for λ 0 = t.
6.5.2.Heuristic argument for a(L).Next we give a heuristic argument in support of [PSTVA21, Conjecture 1.1] (and [BM90, §3.3]) regarding the expected exponent a(L) of B in the asymptotic formula (6.8) for split toric varieties over Q.Up to a positive constant, N m,L (B) is the cardinality S of the set of m i -full positive integers y i for i ∈ {1, . . ., s} that satisfy the conditions y L(σ) ≤ B for all σ ∈ Σ max .We recall that y L(σ) = s i=1 y αi,σ i .One of the ideas of the hyperbola method is to dissect the region of summation for the variables y 1 , . . ., y s , into different boxes.Assume that we consider a box where say y i ∼ B i (here we mean that for example B i ≤ y i ≤ 2B i for i ∈ {1, . . ., s}), and let B i = B ti .What contribution do such vectors y = (y 1 , . . ., y s ) give to computing the cardinality S? First we note that the contribution from this box is box contribution = In order for this to be a box that we count by S, the parameters (t 1 , . . ., t s ) need to satisfy s i=1 t i α i,σ ≤ 1, ∀σ ∈ Σ max (6.15) and t 1 , . . ., t s ≥ 0. (6.16) In order to find the size of S we hence have the following linear programming problem P: Maximize the function s i=1 t i 1 m i (6.17) under the conditions (6.15) and (6.16).The conditions (6.15) and (6.16) define a polytope P in R s and by the theory of linear programming we know that the maximum of the function s i=1 t i 1 mi is obtained on at least one of its vertices.The dual linear programming problem D is given by the following problem: Minimize the function σ∈Σmax λ σ under the conditions (6.10) and (6.11).By the strong duality property in linear programming [Dan98, Chapter 6], both problems have a finite optimal solution and these values are equal.Since a(L) is positive, by Proposition 6.12(i) it is the solution of the dual linear programming problem D and also of P. 6.5.3.Heuristic argument for b(L).Now we give a heuristic argument in support of [PSTVA21, Conjecture 1.1] (and [BM90, §3.3]) regarding the expected exponent a(L) of B in the asymptotic formula (6.8) for split toric varieties over Q.We keep the setting introduced above.
If we cover the region of summation y L(σ) ≤ B for all σ ∈ Σ max by dyadic boxes, then the maximal value of the count is attained on boxes, that are located at the maximal face F of the polytope P where the function in (6.17) is maximized.Working with a dyadic dissection this suggests that the leading term should be of order B a(L) (log B) k , where k is equal to the dimension of the face F .The next proposition shows that k = b(L) − 1.Hence, the heuristic expectation we obtained from the hyperbola method matches the prediction in [PSTVA21, Conjecture 1.1].
We recall from Lemma 6.7 that F ⊆ H, where H is the space of linear functions on Pic(X) R .With this identification, the cone generated by P ∩ H is the space of linear functions on Pic(X) R that are nonnegative on Eff(X) (i.e., the cone in H dual to Eff(X)) by [CLS11, Proposition 1.2.8].Proposition 6.13.The cone generated by F is dual to the minimal face of Eff(X) that contains a(L)[L] + [K X + ∆].In particular, b(L) = dim F + 1.

r i=1 c
bi i where [D] = (b 1 , . . ., b r ) with respect to the fixed basis of Pic(X).Then, as in [Pie16, §2],X(K) = X (O K ) = c∈C r π c (Y c (O K )) ,where π c : Y c → X is the twist of π defined in [FP16, Theorem 2.7].The fibers of π c | Y c (O K ) are all isomorphic to (O × K ) r , and Y c (O K ) ⊆ A s (O K ) is the subset of points y ∈ s i=1 c Di that satisfy σ∈Σmax y Dσ c −Dσ = O K .