On curves with high multiplicity on ℙ(

New York Journal of Mathematics

New York J. Math.27(2021) 1060–1084.

On curves with high multiplicity on ℙ(𝒂, 𝒃, 𝒄) for 𝐦𝐢𝐧(𝒂, 𝒃, 𝒄) ≤ 𝟒

David McKinnon, Rindra Razafy, Matthew Satriano and Yuxuan Sun

Abstract. On a weighted projective surfaceℙ(𝑎, 𝑏, 𝑐)withmin(𝑎, 𝑏, 𝑐) ≤ 4, we compute lower bounds for theeffective thresholdof an ample divisor, in other words, the highest multiplicity a section of the divisor can have at a specified point. We expect that these bounds are close to being sharp. This translates into finding divisor classes on the blowup ofℙ(𝑎, 𝑏, 𝑐)that generate a cone contained in, and probably close to, the effective cone.

Contents

1. Introduction 1060

Acknowledgments 1063

2. Proof of Proposition 1.2 and preliminary reductions 1064 3. Ehrhart quasi-polynomials for𝑏𝐷_𝑥and𝑐𝐷_𝑥 1068 4. Bounding𝑐₀(𝑏𝐷_𝑥, 𝑛)and𝑐₀(𝑐𝐷_𝑥, 𝑛) 1072

5. Proof of Theorem 1.5 1077

References 1083

1. Introduction

Given a projective variety𝑋and a point𝑄 ∈ 𝑋, it is, in general, a notoriously difficult problem to calculate the pseudo-effective cone of the blow-up Bl𝑄(𝑋) in terms of the pseudo-effective cone of𝑋. Even addressing thea priorieas- ier question of when Bl𝑄(𝑋)is a Mori Dream Space, where𝑋 = ℙ(𝑎, 𝑏, 𝑐)is a weighted projective surface and𝑄is the identity of its torus, is already challeng- ing and has a rich history [Hun82, Cut91, Sri91, GNW94, CT15, GK16, He17, GGK20]. To gain information about the pseudo-effective cone of Bl𝑄(𝑋), we consider the following quantity, cf. [Fuj92].

Received November 23, 2020.

2010Mathematics Subject Classification. 14C15, 14F43.

Key words and phrases. weighted projective spaces, curves, Ehrhart polynomial, multiplicity.

The first author and third authors were partially supported by a Discovery Grant from the Na- tional Science and Engineering Board of Canada. The second and fourth authors were supported by an Undergraduate Student Research Award from the National Science and Engineering Board of Canada.

ISSN 1076-9803/2021

1060

Definition 1.1. Let 𝑋 be a projective variety defined over a field𝑘, 𝐷 a 𝑘- rationalℚ-divisor, and 𝑄 a 𝑘-rational point of𝑋. Let 𝜋 be the blowup of 𝑋 at𝑄and𝐸the exceptional divisor of𝜋. We say theeffective thresholdis

𝛾_𝑄(𝐷) ∶= sup{𝛾 > 0 ∣ 𝜋^∗(𝐷) − 𝛾𝐸 is pseudo-effective}.

The quantity𝛾_𝑄(𝐷)can be reinterpreted concretely as follows:if there is a divisor in the class of𝐷with multiplicity𝑚at𝑄, then𝛾_𝑄(𝐷) ≥ 𝑚. Conversely, if𝛾_𝑄(𝐷) = 𝑚, then for all𝜖 > 0, the class𝜋^∗𝐷 − (𝑚 − 𝜖)𝐸is pseudo-effective, so 𝐷contains divisors of multiplicity arbitrarily close to𝑚, at least in aℚ-divisor sense. So, computing𝛾_𝑄(𝐷)essentially amounts to computing

sup

𝐶,𝑚

{1

𝑚mult_𝑄(𝐶)}

as𝑚varies through positive integers and𝐶varies through divisors in the divisor class𝑚𝐷.

In this paper, we give characteristic-free lower bounds for𝛾_𝑄(𝐷)in the case where𝑋is the weighted projective surfaceℙ(𝑎, 𝑏, 𝑐)andmin(𝑎, 𝑏, 𝑐) ≤ 4. In fact, we do more than this: we introduce a combinatorial quantity 𝛾_expected which is a lower bound on𝛾, and compute𝛾_expectedexactly. It is worth remark- ing here that although the motivation for studying𝛾_𝑄 is geometric, our lower bounds on 𝛾_𝑄 also have consequences for Diophantine approximation prob- lems related to generalizations of Roth’s famous 1955 theorem [Ro55], see e.g., [MR16, Theorem 3.3] and [MS20, Section 8].

In [GGK20], the authors make a series of detailed calculations closely related to what we compute in this paper. In particular, they search the spaces of global sections of toric surfaces of Picard rank one for irreducible curves whose strict transforms have negative self-intersection upon blowing up a point. If there is such a curve, then the pseudo-effective cone of the blowup will be finitely gen- erated by the exceptional divisor of the blowup and another curve of negative self-intersection.

In this paper, we compute not only curves, but also the corresponding value of the effective threshold. We do not prove that the curves we find are always generators of the pseudo-effective cone, but in most cases the value of𝛾 we compute is expected to be equal or very close to the actual value. As the authors of [GGK20] also point out, our quantity𝛾_expectedis expected to be very close to the actual value of𝛾.

Since𝑋 = ℙ(𝑎, 𝑏, 𝑐)is a toric surface, if the point𝑄does not lie in the main torus orbit𝑇, then computing𝛾_𝑄 is generally straightforward, so we may as- sume that𝑄lies in𝑇. Furthermore, we can choose𝑎,𝑏, and𝑐to be pairwise coprime, with𝑎 ≤ 𝑏 ≤ 𝑐. These inequalities are always strict unless𝑎 = 𝑏 = 1, in which case𝛾can be computed directly. Thus, we may assume that𝑎 < 𝑏 < 𝑐. Finally, since𝑋has Picard rank1, it suffices to compute𝛾_𝑄(𝐻), where𝐻is the generator of the Cartier class group.

Our first result concerns the case𝑎 < 4and serves as a warm-up to our main result.

Proposition 1.2. Let𝑎 < 𝑏 < 𝑐be pairwise coprime, so we may write𝑐 = 𝑝𝑎+𝑞𝑏 with𝑝, 𝑞 ∈ ℤand0 ≤ 𝑞 < 𝑎. Let𝑄be in the torus ofℙ(𝑎, 𝑏, 𝑐)and𝐻be the generator of the Cartier class group. Then

𝛾_𝑄(𝐻) ≥ {(𝑞 + 1)𝑏, 𝑝 ≥ 0

(𝑎 − 1)𝑏, 𝑝 < 0and𝑎 ≤ 3.

We do not claim that this Proposition is new. Indeed, in [HKL18], Hausen, Keicher, and Laface prove a number of results along these lines, and moreover they obtain all the results of Proposition 1.2, as a consequence of their Theorems 1.1 and 1.2. Despite the lack of novelty in Proposition 1.2, we present a proof of it to illustrate our techniques in a simpler setting.

Proposition 1.2 yields lower bounds on𝛾_𝑄when𝑎 ≤ 3. Moving from𝑎 ≤ 3to 𝑎 = 4is significantly more involved. In order to state our results, we first discuss our main technique of proof. Note that if𝑚 ∈ ℚ⁺and𝑚𝐻is a Weil divisor such thatℎ⁰(𝑚𝐻) >(_𝜈+1

2

)

, then there is a global section𝑔of𝑚𝐻that vanishes at𝑄 to order𝜈. Writing𝑚 = ^𝑚1

𝑚₂ with𝑚₁, 𝑚₂ ∈ ℤ⁺, we see 𝑔^𝑚2 ∈ 𝐻⁰(𝑋, 𝑚₁𝐻) vanishes to order𝜈𝑚₂. By definition, it follows that𝛾_𝑄(𝐻) ≥ ^𝜈𝑚2

𝑚₁ = ^𝜈

𝑚. This motivates the following definition.

Definition 1.3. For any Weil divisor𝐷, let

𝜈(𝐷) ∶= max {𝑑 ∈ ℤ⁺∣ ℎ⁰(𝐷) >(𝑑 + 1 2

) } .

If𝐻denotes the generator of the Cartier class group ofℙ(𝑎, 𝑏, 𝑐)with𝑎 < 𝑏 < 𝑐, then let

𝛾_expected(𝐻) ∶= sup {𝜈(𝑚𝐻)

𝑚 ∣ 𝑚 ∈ 1

𝑏ℤ⁺∪ 1 𝑐ℤ⁺} .

Remark1.4. Note that the definition of𝛾_expectedconsiders only some of the Weil divisors. In particular, since the Weil class group ofℙ(𝑎, 𝑏, 𝑐)is generated by

1

𝑎𝑏𝑐𝐻, every Weil divisor that appears in the definition of𝛾_expectedis a multiple of𝑎in the Weil class group.

We can now state our main result. Recall that Proposition 1.2 already yields lower bounds on𝛾_𝑄when𝑝 ≥ 0in general, so we turn to the case𝑝 < 0. Theorem 1.5. With notation and hypotheses as in Proposition1.2, assume𝑎 = 4 and𝑝 < 0. Then

𝛾_𝑄(𝐻) ≥ 𝛾_expected(𝐻) = 𝜈(𝐷₀) 𝑚₀ ,

where𝐷₀ ∼ 𝑚₀𝐻 and𝜈(𝐷₀)are computed exactly as follows. Given our con- straints, we have2 < ^𝑏

−𝑝 < ¹⁶

3. Divide the interval[2,¹⁶

3]into a countably infinite sequence of intervals of the form

𝐼_𝑘 ∶= [ 16(𝑘 + 1)²

8(𝑘 + 1)²− 4(𝑘 + 1) − 1, 16𝑘² 8𝑘²− 4𝑘 − 1]

with𝑘 ∈ ℤ⁺. Then the class of𝐷₀is given as follows, depending on the value of

𝑏

−𝑝 ∈ 𝐼_𝑘: (1) If ^𝑏

−𝑝 ∈ 𝐼_𝑘,−^′ ∶= [ ^16(𝑘+1)

2

8(𝑘+1)²−4(𝑘+1)−1,^2𝑘+1

𝑘 ], then𝐷₀ ∼ ^2𝑘+3

𝑐 𝐻with𝜈(𝐷₀) = 4(𝑘 + 1).

(2) If ^𝑏

−𝑝 ∈ 𝐼_𝑘,+^′ ∶= [^2𝑘+1

𝑘 , ^4(2𝑘+1)

2

8𝑘²+4𝑘−1], then𝐷₀ ∼ ^2𝑘+1

𝑏 𝐻with𝜈(𝐷₀) = 4(𝑘+1).

(3) If ^𝑏

−𝑝 ∈ 𝐼_𝑘,−^′′ ∶= [^4(2𝑘+1)

2

8𝑘²+4𝑘−1, ^4𝑘

2𝑘−1], then𝐷₀∼ ^𝑘+1

𝑐 𝐻with𝜈(𝐷₀) = 2𝑘 + 1.

(4) If ^𝑏

−𝑝 ∈ 𝐼_𝑘,+^′′ ∶= [ ^4𝑘

2𝑘−1, ^16𝑘

2

8𝑘²−4𝑘−1], then𝐷₀∼ ^𝑘

𝑏𝐻with𝜈(𝐷₀) = 2𝑘 + 1.

Remark1.6. The quantity𝛾_expected(𝐻)is the lower bound for𝛾(𝐻)obtained by simple linear algebra: the vanishing to order𝑛of a section of𝐻is equivalent to the vanishing of(_𝑛+1

2

)

linear forms on the space of sections of𝐻. We therefore have𝛾(𝐻) ≥ 𝛾_expected(𝐻)trivially.

However, one also expects that the two quantities are not so different. First of all, the divisibility restriction in𝛾_{𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑}does not exclude any Cartier divisors, and in the examples that we are aware of, does not change the value of𝛾at all.

More significantly, if 𝛾(𝐻) > 𝛾_expected(𝐻) at some point 𝑄 in the main torus orbit, then there is a section𝑠of some multiple𝑚𝐻of𝐻that has an order of vanishing that is greater than𝜈(𝑚𝐻)at𝑄. For any element𝜎of the torus, the section𝜎(𝑠)has unusually high order of vanishing at𝜎(𝑄), so foreverypoint of the main orbit, there is a section of𝑚𝐻 that has unusually high order of vanishing there. This is unlikely – though not downright impossible – and so one expects the two quantities to be close.

Nevertheless, there are examples where𝛾and𝛾_expecteddo not agree. For ex- ample, if(𝑎, 𝑏, 𝑐) = (5, 33, 49)or(8, 15, 43), Kurano and Matsuoka ([KM09]) showed that𝛾 and 𝛾_expected are not the same. Several other authors, includ- ing Gonzalez Anaya, Gonzalez, and Karu, have obtained other very interest- ing results along these lines, of which an excellent summary can be found in [GAGK21].

The rest of the paper is organized as follows. Section 2 proves Proposition 1.2 and describes some preliminary reductions for Theorem 1.5. Section 3 com- putes the main terms in the count of global sections of multiples of𝐻. Section 4 then begins the process of bounding the error terms, and Section 5 finishes the proof of Theorem 1.5.

Acknowledgments

We are grateful to Kalle Karu for many enlightening email exchanges. We thank the anonymous referee for helpful suggestions. This paper is the outcome of an NSERC-USRA project; we thank NSERC for their support.

2. Proof of Proposition 1.2 and preliminary reductions

Throughout this paper, we let𝑥, 𝑦, and𝑧 be the weighted projective coor- dinates onℙ(𝑎, 𝑏, 𝑐)with weights𝑎,𝑏, and𝑐, respectively. We let𝐷_𝑥,𝐷_𝑦, and 𝐷_𝑧 denote the Weil divisors defined by the vanishing of 𝑥, 𝑦, and𝑧, respec- tively. We let 𝐻 denote the generator of the Cartier class group, so we have 𝐻 ∼ 𝑏𝑐𝐷_𝑥 ∼ 𝑎𝑐𝐷_𝑦 ∼ 𝑎𝑏𝐷_𝑧. Given any Weil divisor𝐷, we let𝑃_𝐷 ⊂ ℝ²be the associated polytope with the property thatℎ⁰(𝐷) = |𝑃_𝐷 ∩ ℤ²|. We sometimes abusively denote|𝑃_𝐷 ∩ ℤ²|by|𝑃_𝐷|.

It will frequently be useful to write our divisors as integer multiples of𝐷_𝑥. Note that if𝛿 ∈ {𝑏, 𝑐}and𝑚 ∈ ¹

𝛿ℤ⁺, then𝑚𝐻 ∼ 𝑛𝛿𝐷_𝑥where𝑛 = ^𝑏𝑐𝑚

𝛿 ∈ ℤ⁺. After a preliminary lemma, we prove Proposition 2.2 which is a slightly more general version of Proposition 1.2.

Lemma 2.1. With notation and hypotheses as in Proposition1.2, if𝑝 < 0, then 𝑞 ≠ 1. In particular, if𝑝 < 0and𝑎 ≤ 3, then𝑎 = 3and𝑞 = 2.

Proof. If𝑞 = 1, then𝑐 = 𝑝𝑎 + 𝑏 < 𝑏which is a contradiction. If𝑝 < 0and 𝑎 ≤ 3, then𝑞 = 0or𝑞 ≥ 2. The former case cannot occur as it implies𝑐 = 𝑝𝑎 and hence𝑝 > 0. The latter case implies2 ≤ 𝑞 ≤ 𝑎 − 1so𝑞 = 2and𝑎 = 3. Proposition 2.2. With notation and hypotheses as in Proposition1.2, we have

𝛾_𝑄(𝐻) ≥ {(𝑞 + 1)𝑏, 𝑝 ≥ 0

(𝑎 − 1)𝑏, 𝑝 < 0, 𝑞 = 𝑎 − 1, and ^−𝑝𝑎

𝑏 ≤ 1.

Furthermore, if𝑎 ≤ 3and𝑝 < 0, then𝑞 = 𝑎 − 1and ^−𝑝𝑎

𝑏 ≤ 1automatically hold.

Proof. Note that since𝑎,𝑏, and𝑐are pairwise coprime,𝑝 ≠ 0. First suppose 𝑝 > 0. Then the polytope𝑃_𝑎𝐷𝑧 is the convex hull of0,(𝑞, −𝑎), and(^−𝑝𝑎

𝑏 , −𝑎), so it contains the triangle𝑇with vertices0,(𝑞, −𝑎),(0, −𝑎). By Pick’s Theorem, 1 +(_𝑞+2

2

) ≤ |𝑇 ∩ ℤ²| ≤ |𝑃_𝑎𝐷𝑧 ∩ ℤ²|, which implies𝜈(𝑎𝐷_𝑧) ≥ 𝑞 + 1. Since 𝑎𝐷_𝑧∼ ¹

𝑏𝐻, we find𝛾_𝑄(𝐻) ≥ (𝑞 + 1)𝑏. Next, suppose𝑝 < 0,𝑞 = 𝑎 − 1, and ^−𝑝𝑎

𝑏 ≤ 1. Notice that the polytope𝑃_𝑎𝐷𝑧 is given by the vertices as above, and it contains the triangle 𝑇with vertices 0, (𝑞, −𝑎), and(1, −𝑎)by 𝑝 < 0and ^−𝑝𝑎

𝑏 ≤ 1. From 𝑞 = 𝑎 − 1 and Pick’s Theorem, we have 1 +(_𝑎

2

) = |𝑇 ∩ ℤ²|, which, as in the previous paragraph, implies𝛾_𝑄(𝐻) ≥ (𝑎 − 1)𝑏.

Finally, we note that if𝑎 ≤ 3and𝑝 < 0, then Lemma 2.1 tells us𝑎 = 3and 𝑞 = 2 = 𝑎 − 1. Then,𝑏 < 𝑐 = 𝑝𝑎 + 2𝑏implies^−𝑝𝑎

𝑏 < 1.

The rest of the paper is concerned with the proof of Theorem 1.5. By Lemma 2.1, since𝑝 < 0and𝑎, 𝑏, 𝑐are pairwise coprime, we must have

𝑞 = 3 = 𝑎 − 1.

We begin by analyzing𝜈(𝐷₀).

Proposition 2.3. With notation and hypotheses as in Theorem1.5, if ^𝑏

−𝑝 ∈ 𝐼^′

𝑘 ∶=

𝐼^′

𝑘,+∪ 𝐼^′

𝑘,−, resp.𝐼^′′

𝑘 ∶= 𝐼^′′

𝑘,+∪ 𝐼^′′

𝑘,−, and𝐷₀is as in the conclusion of the theorem, then𝜈(𝐷₀) ≥ 4(𝑘 + 1), resp.2𝑘 + 1.

Proof. Let𝑛₀ ∈ ℤbe such that𝐷₀ ∼ ^𝑛0

𝑏𝐻 ∼ 𝑛₀𝑐𝐷_𝑥or𝐷₀∼ ^𝑛0

𝑐 𝐻 ∼ 𝑛₀𝑏𝐷_𝑥. In the former (respectively latter) case,ℎ⁰(𝐷₀)is given by the number of integer lattice points lying in the polytope

𝑃_𝑛

0𝑐𝐷_𝑥 = Conv ((0, 0), (−𝑛₀𝑐

𝑏, 0) , (−3𝑛₀, 4𝑛₀)) respectively

𝑃_𝑛

0𝑏𝐷_𝑥 = Conv ((0, 0), (−𝑛₀, 0), (−3𝑛₀𝑏 𝑐, 4𝑛₀𝑏

𝑐)) . First consider the case _−𝑝^𝑏 ∈ 𝐼^′_𝑘. As in Theorem 1.5,𝐼^′_𝑘,+ ∶= [^2𝑘+1

𝑘 , ^4(2𝑘+1)

2

8𝑘²+4𝑘−1] and𝐼^′_𝑘,− ∶= [ ^16(𝑘+1)

2

8(𝑘+1)²−4(𝑘+1)−1,^2𝑘+1

𝑘 ]. If ^𝑏

−𝑝 ∈ 𝐼_𝑘,+^′ , then𝐷₀ ∼ 𝑛₀𝑐𝐷_𝑥with𝑛₀ = 2𝑘 + 1and if ^𝑏

−𝑝 ∈ 𝐼^′

𝑘,−, then𝐷₀∼ 𝑛₀𝑏𝐷_𝑥with𝑛₀= 2𝑘 + 3. Let 𝑃^′= Conv((0, 0), (−(2𝑘 + 3), 0), (−3(2𝑘 + 1), 4(2𝑘 + 1))).

We then have𝑃^′ ⊂ 𝑃. Indeed, if ^𝑏

−𝑝 ∈ 𝐼^′

𝑘,+, then the inclusion follows from

−(2𝑘 + 1)^𝑐

𝑏 = −(2𝑘 + 1)(4^𝑝

𝑏 + 3) ≤ −(2𝑘 + 1)(4 ^−𝑘

2𝑘+1 + 3) = −(2𝑘 + 3). If

𝑏

−𝑝 ∈ 𝐼^′_𝑘,−, then the inclusion follows from−3(2𝑘 + 3)^𝑏

𝑐 ≤ −3(2𝑘 + 1)and 4(2𝑘 + 3)^𝑏

𝑐 ≥ 4(2𝑘 + 1). So, in either case, we have

|𝑃^′∩ ℤ²| ≤ ℎ⁰(𝐷₀).

Note that the area of𝑃^′is given by 𝐴(𝑃^′) = 1

2(4(2𝑘 + 1)(2𝑘 + 3)) = 2(2𝑘 + 1)(2𝑘 + 3) and the number of lattice points on its boundary is given by

𝐵(𝑃^′) = (2𝑘 + 3) + (2𝑘 + 1) + 4 = 4𝑘 + 8.

Since𝑃^′is a lattice polygon, applying Pick’s Theorem, we have

|𝑃^′∩ ℤ²| = 𝐴(𝑃^′) + 1

2𝐵(𝑃^′) + 1 = 8𝑘²+ 18𝑘 + 11 =(4(𝑘 + 1) + 1 2

) + 1, which shows𝜈(𝐷₀) ≥ 𝜈(𝑃^′) = 4(𝑘 + 1).

For ^𝑏

−𝑝 ∈ 𝐼^′′_𝑘, the same proof works when using

𝑃^′′ = Conv((0, 0), (−(𝑘 + 1), 0), (−3𝑘, 4𝑘))

in place of𝑃^′.

Proposition 2.3 therefore gives the lower bound 𝛾_expected(𝐻) ≥ 𝜈(𝐷₀)

𝑚₀ , (2.4)

where𝐷₀= 𝑚₀𝐻is the divisor class described in Theorem 1.5. To obtain upper bounds, we introduce the following quantities and make use of the subsequent lemma. Let

𝛾_{expected,𝑏}(𝐻) ∶= sup {1

𝑚𝜈(𝑚𝐻) ∣ 𝑚 ∈ 1 𝑐ℤ⁺} 𝛾_{expected,𝑐}(𝐻) ∶= sup {1

𝑚𝜈(𝑚𝐻) ∣ 𝑚 ∈ 1 𝑏ℤ⁺} .

Then, we may bound 𝛾_expected(𝐻)from above by bounding𝛾_{expected,𝑏}(𝐻) and 𝛾_{expected,𝑐}(𝐻)from above, given that

𝛾_expected(𝐻) = max{

𝛾_{expected,𝑏}(𝐻), 𝛾_{expected,𝑐}(𝐻)}

. (2.5)

Lemma 2.6. Supposeℙ(4, 𝑏₀, 𝑐₀),ℙ(4, 𝑏_𝐿, 𝑐_𝐿), andℙ(4, 𝑏_𝑈, 𝑐_𝑈)satisfy the hy- potheses of Theorem1.5, except we need not assume𝑝 < 0. Suppose ^𝑏𝐿

−𝑝_𝐿 < ^𝑏0

−𝑝₀ <

𝑏_𝑈

−𝑝_𝑈 and let𝐻₀,𝐻_𝐿, and𝐻_𝑈 denote the generators of the respective Cartier class groups. Then

1

𝑐₀𝛾_{expected,𝑏0}(𝐻₀) ≤ 1

𝑐_𝐿𝛾_{expected,𝑏𝐿}(𝐻_𝐿)

and 1

𝑐₀𝛾_{expected,𝑐0}(𝐻₀) ≤ 1

𝑐_𝑈𝛾_{expected,𝑐𝑈}(𝐻_𝑈).

Proof. Since𝑐 = 𝑝𝑎+𝑞𝑏 = 4𝑝+3𝑏, we see^𝑏

𝑐 = ¹

4^𝑝

𝑏+3. As a result,^𝑏𝑈

𝑐_𝑈 < ^𝑏0

𝑐₀ < ^𝑏𝐿

𝑐_𝐿. It follows that

𝑃₀∶= Conv ((0, 0), (−1, 0), (−3𝑏₀ 𝑐₀, 4𝑏₀

𝑐₀))

⊂ Conv ((0, 0), (−1, 0), (−3𝑏_𝐿 𝑐_𝐿, 4𝑏_𝐿

𝑐_𝐿))

=∶ 𝑃_𝐿.

Since𝑃₀, resp.𝑃_𝐿, is the polytope of 𝑏𝐷_𝑥 onℙ(4, 𝑏₀, 𝑐₀), resp.ℙ(4, 𝑏_𝐿, 𝑐_𝐿), we have𝜈(𝑛𝑃₀) ≤ 𝜈(𝑛𝑃_𝐿)for all𝑛 ≥ 1, and so

(1∕𝑐₀)𝛾_{expected,𝑏}(𝐻₀) ≤ (1∕𝑐_𝐿)𝛾_{expected,𝑏𝐿}(𝐻_𝐿).

We obtain the inequality(1∕𝑐₀)𝛾_{expected,𝑐}(𝐻₀) ≤ (1∕𝑐_𝑈)𝛾_{expected,𝑐𝑈}(𝐻_𝑈)in a similar manner from the inclusion

Conv ((0, 0), (−𝑐₀

𝑏₀, 0) , (−3, 4)) ⊂ Conv ((0, 0), (−𝑐_𝑈

𝑏_𝑈, 0) , (−3, 4)) , the left-hand, resp. right-hand, side being the polytope of𝑐𝐷_𝑥 onℙ(4, 𝑏₀, 𝑐₀),

resp.ℙ(4, 𝑏_𝑈, 𝑐_𝑈).

Remark 2.7. Lemma 2.6 may be used to reduce the proof of Theorem 1.5 to special classes of weighted projective spaces with desirable arithmetic proper- ties. The key idea is to compare the values of𝛾_{expected,𝑏}(𝐻)and𝛾_{expected,𝑐}(𝐻)on different weighted projective spaces to generate upper bounds.

Fix a weighted projective spaceℙ(4, 𝑏, 𝑐)satisfying the hypotheses of Theo- rem 1.5 and let𝐷₀∼ 𝑚₀𝐻and𝜈₀ ∶= 𝜈(𝐷₀)be as predicted by theorem. We will suppose that𝑚₀= ^𝑛0

𝑐 with𝑛₀∈ ℤ⁺(the case where𝑚₀∈ ¹

𝑏ℤ⁺is handled sim- ilarly). We must prove𝛾_expected(𝐻) = ^𝜈0

𝑚₀ = ^𝑐𝜈0

𝑛₀. Let𝐼 = [^𝛽1

𝛼₁,^𝛽2

𝛼₂]be an interval of the form𝐼_𝑘,±^′ or𝐼_𝑘,±^′′ as in Proposition 2.3, where ^𝑏

−𝑝 ∈ 𝐼. Assume ^𝑏

−𝑝is in the interior of𝐼. We must show𝛾_expected(𝐻) = ^𝑐𝜈0

𝑛₀. By (2.5), this is equivalent to proving

𝛾_{expected,𝑏}(𝐻) ≤ 𝑐𝜈₀

𝑛₀ and 𝛾_{expected,𝑐}(𝐻) ≤ 𝑐𝜈₀ 𝑛₀.

This may be done as follows: fix increasing sequences of positive integers{𝑏_𝑖}_𝑖 and {−𝑝_𝑖}_𝑖 for which 𝛼₁𝑏_𝑖 − 𝛽₁(−𝑝_𝑖) = 1 and𝑐_𝑖 ∶= 4𝑝_𝑖 + 3𝑏_𝑖 > 𝑏_𝑖 is such that 4,𝑏_𝑖,𝑐_𝑖 are pairwise coprime. We may always find such sequences, since 𝛼₁, 𝛽₁ > 0are coprime in all cases listed in Theorem 1.5. Let𝐻_𝑖 denote the generator of the Cartier class group ofℙ(4, 𝑏_𝑖, 𝑐_𝑖). Then for𝑖sufficiently large,

𝑏_𝑖

−𝑝_𝑖 ∈ 𝐼is monotonically decreasing with ^𝑏𝑖

−𝑝_𝑖 → ^𝛽1

𝛼₁. Given ^𝑏

−𝑝 ∈ 𝐼, there exists an𝑁large enough such that ^𝑏𝑁

−𝑝_𝑁 < ^𝑏

−𝑝, so by Lemma 2.6, 1

𝑐𝛾_{expected,𝑏}(𝐻) ≤ 1

𝑐_𝑁𝛾_{expected,𝑏}

𝑁(𝐻_𝑁) ≤ 1

𝑐_𝑁+1𝛾_{expected,𝑏}

𝑁+1(𝐻_𝑁+1) ≤ … (2.8) Similarly, fix increasing sequences of positive integers {𝑏_𝑖^′}_𝑖 and {−𝑝^′_𝑖}_𝑖 for which𝛼₂𝑏^′_𝑖 − 𝛽₂(−𝑝^′_𝑖) = −1and 𝑐_𝑖^′ ∶= 4𝑝_𝑖^′ + 3𝑏^′_𝑖 > 𝑏_𝑖^′ is such that 4,𝑏^′_𝑖, 𝑐^′_𝑖 are pairwise coprime. As above, such sequences always exist. Let𝐻_𝑖^′ denote the generator of the Cartier class group of ℙ(4, 𝑏_𝑖^′, 𝑐_𝑖^′). Then for 𝑖 sufficiently large, ^𝑏

′ 𝑖

−𝑝_𝑖^′ ∈ 𝐼is monotonically increasing with ^𝑏

′ 𝑖

−𝑝^′_𝑖 → ^𝛽2

𝛼₂. Choosing an𝑁large enough such that ^𝑏

−𝑝 < ^𝑏

′𝑛

−𝑝^′_𝑛, we have by Lemma 2.6 1

𝑐𝛾_{expected,𝑐}(𝐻) ≤ 1

𝑐_𝑁^′ 𝛾_{expected,𝑐}^′

𝑁(𝐻_𝑁^′ ) ≤ 1

𝑐^′_𝑁+1𝛾_{expected,𝑐}^′

𝑁+1(𝐻_𝑁+1^′ ) ≤ … (2.9) We claim that to prove Theorem 1.5 forℙ(4, 𝑏, 𝑐), it is enough to show

𝜈₀ 𝑛₀ ≥ 1

𝑛𝜈 (𝑛

𝑐_𝑖𝐻_𝑖) (2.10)

and

𝑐^′_𝑖𝜈₀ 𝑛₀ > 𝑏^′_𝑖

𝑛𝜈 (𝑛

𝑏_𝑖𝐻^′_𝑖) (2.11)

for all𝑖sufficiently large and all𝑛 ∈ ℤ⁺. Indeed, (2.10) implies 𝑐_𝑖𝜈₀

𝑛₀ ≥ 𝑐_𝑖sup {1 𝑛𝜈 (𝑛

𝑐_𝑖𝐻_𝑖) ∣ 𝑛 ∈ ℤ⁺}

= sup {1

𝑚𝜈 (𝑚𝐻_𝑖) ∣ 𝑚 ∈ 1 𝑐_𝑖ℤ⁺}

= 𝛾_{expected,𝑏}

𝑖(𝐻_𝑖)

for all𝑖sufficiently large, which when combined with (2.8) shows 𝛾_{expected,𝑏}(𝐻) ≤ 𝑐𝜈₀

𝑛₀. Similarly, (2.11) implies

𝑐^′_𝑖𝜈₀

𝑛₀ > 𝛾_{expected,𝑐}^′

𝑖(𝐻^′_𝑖), which when combined with (2.9) shows

𝛾_{expected,𝑐}(𝐻) < 𝑐𝜈₀ 𝑛₀ .

which is exactly our goal. Further, the strict inequality in this latter case will allow us to conclude that no𝐷of the form ^𝑛_𝑐𝐻(other than the case where𝑛is divisible by𝑛₀) gives the required upper bound for𝛾_expected(𝐻). Together with the lower bound (2.4), this computes𝛾_expected(𝐻).

Note that the above computes𝛾_expected(𝐻)assuming ^𝑏

−𝑝is in the interior of𝐼. If ^𝑏

−𝑝 is an endpoint of𝐼, then it is straightforward to check that𝑏 = 2𝑘 + 1and

−𝑝 = 𝑘, in which case𝑐 = 2𝑘 + 3. In that case, one may verify Theorem 1.5 directly using that the Erhart quasi-polynomial computed in Section 3 is an actual polynomial.

3. Ehrhart quasi-polynomials for𝒃𝑫_𝒙 and𝒄𝑫_𝒙

Our first goal in this section is to give an expression for the number of lattice points in the polytopes𝑃_𝑛𝑏𝐷

𝑥 and𝑃_𝑛𝑐𝐷

𝑥.

Proposition 3.1. Keep the notation and hypotheses of Theorem1.5, let𝛿 ∈ {𝑏, 𝑐}, and set𝑠 = ^𝑏

𝑐. Then

|𝑃_𝑛𝛿𝐷

𝑥 ∩ ℤ²| = 𝑐₂(𝛿𝐷_𝑥, 𝑛)𝑛²+ 𝑐₁(𝛿𝐷_𝑥, 𝑛)𝑛 + 𝑐₀(𝛿𝐷_𝑥, 𝑛), where the𝑐_𝑖’s are given as follows.

(1) For𝛿 = 𝑏, we have𝑐₂(𝑏𝐷_𝑥, 𝑛) = 2𝑠,𝑐₁(𝑏𝐷_𝑥, 𝑛) = ¹

2(1 + 𝑠 + ⁴

𝑐), and

𝑐₀(𝑏𝐷_𝑥, 𝑛) = 1 − 1 8𝑠

(

{4𝑠𝑛}²− {4𝑠𝑛}

)

−5 2{𝑠𝑛} +

4 {⌊4𝑠𝑛⌋

4

}

∑

𝑗=0

{3 4𝑗}

+𝑏 − 1 2 {4𝑛

𝑐 } −

𝑏 {⌊4𝑠𝑛⌋

𝑏

}

∑

𝑗=0

{−𝑝 𝑏 𝑗} .

(2) For𝛿 = 𝑐, we have𝑐₂(𝑐𝐷_𝑥, 𝑛) = ²

𝑠,𝑐₁(𝑐𝐷_𝑥, 𝑛) =¹

2(1 + ¹

𝑠+ ⁴

𝑏), and

𝑐₀(𝑐𝐷_𝑥, 𝑛) = 1 − {4𝑛

𝑏 } −𝑏 − 1 2 {4𝑛

𝑏 } +

𝑏{^4𝑛

𝑏}

∑

𝑗=0

{−𝑝 𝑏 𝑗} . Proof. First consider𝛿 = 𝑏.

|𝑃_𝑛𝑏𝐷

𝑥 ∩ ℤ²| ∶= |||Conv(𝐴, 𝐵, 𝐶)||| ∶=

||||

|||

Conv ((0, 0), (−𝑛, 0), (−3𝑛𝑏 𝑐, 4𝑛𝑏

𝑐))||||||| where𝐵𝐶 is given by𝑦 = ^𝑏

𝑝𝑥 + ^𝑏

𝑝𝑛 and𝐴𝐶 is given by𝑦 = −⁴

3𝑥. We will compute|𝑃_𝑛𝑏𝐷

𝑥 ∩ ℤ²|by counting the number of lattice points lying on each line segment𝐴_𝑗𝐵_𝑗, where𝐴_𝑗 = (−³

4𝑗, 𝑗)lies on𝐴𝐶and𝐵_𝑗 = (^𝑝

𝑏𝑗 − 𝑛, 𝑗)lies on𝐵𝐶, for𝑗 = 0, 1, … , ⌊4𝑠𝑛⌋. Here, our approach is similar to that of [L11, Theorem 3.1]. Denote𝑀 = ⌊4𝑠𝑛⌋ = 4𝑠𝑛 − {4𝑠𝑛}. Then,

|𝑃_{𝑛𝑏𝐷𝑥}∩ ℤ²|

=

∑𝑀 𝑗=0

(

⌊

−3 4𝑗

⌋

−⌈ 𝑝 𝑏𝑗 − 𝑛

⌉ + 1)

= (𝑛 + 1)(𝑀 + 1) +

∑𝑀 𝑗=0

(−

⌈3 4𝑗

⌉

+⌊ −𝑝 𝑏 𝑗

⌋ )

= (𝑛 + 1)(𝑀 + 1) +

∑𝑀 𝑗=0

(− (3

4𝑗 + 1 − {3

4𝑗}) +−𝑝

𝑏 𝑗 − {−𝑝 𝑏 𝑗}) +

⌊𝑀 4

⌋ + 1

= 𝑛(𝑀 + 1) +𝑀 4 − {𝑀

4 } + 1 +

∑𝑀 𝑗=0

(−𝑝 𝑏 −3

4) 𝑗 +

∑𝑀 𝑗=0

{3 4𝑗} −

∑𝑀 𝑗=0

{−𝑝 𝑏 𝑗} .

Rewrite the sums involving fractional parts as sums of a linear term in𝑛and a 𝑐−periodic term in𝑛:

∑𝑀 𝑗=0

{3 4𝑗} =

⌊𝑀 4

⌋∑³

𝑗=0

{3 4𝑗} +

4 {𝑀

4

}

∑

𝑗=0

{3 4𝑗} = 3

2(𝑠𝑛 − {𝑠𝑛}) +

4 {⌊4𝑠𝑛⌋

4

}

∑

𝑗=0

{3 4𝑗} ,

∑𝑀 𝑗=0

{−𝑝 𝑏 𝑗} =

⌊𝑀 𝑏

⌋^𝑏−1∑

𝑗=0

{−𝑝 𝑏 𝑗} +

𝑏 {𝑀

𝑏

}

∑

𝑗=0

{−𝑝

𝑏 𝑗} = 𝑏 − 1 2 (4𝑛

𝑐 − {4𝑛 𝑐 }) +

𝑏 {⌊4𝑠𝑛⌋

𝑏

}

∑

𝑗=0

{−𝑝 𝑏 𝑗} ,

where we have used the identity

𝑏−1∑

𝑗=0

{−𝑝

𝑏 𝑗} = 𝑏 − 1 2

given that𝑏and𝑝are coprime. Moreover, by a direct computation,

∑𝑀 𝑗=0

(−𝑝 𝑏 −3

4) 𝑗 = −(𝑀 + 1 2

) 𝑐

4𝑏 = −2𝑠𝑛²+({4𝑠𝑛} −1 2) 𝑛−1

8𝑠 (

{4𝑠𝑛}²− {4𝑠𝑛}

) .

Thus, we can write |𝑃_{𝑛𝑏𝐷𝑥} ∩ ℤ²| = 𝑐₂(𝑏𝐷_𝑥, 𝑛)𝑛² + 𝑐₁(𝑏𝐷_𝑥, 𝑛)𝑛 + 𝑐₀(𝑏𝐷_𝑥, 𝑛), where

𝑐₂(𝑏𝐷_𝑥, 𝑛) = 2𝑠, 𝑐₁(𝑏𝐷_𝑥, 𝑛) =1

2(1 + 𝑠 + 4 𝑐) , 𝑐₀(𝑏𝐷_𝑥, 𝑛) = 1 − 1

8𝑠 (

{4𝑠𝑛}²− {4𝑠𝑛}

)

−5 2{𝑠𝑛}

+

4 {⌊4𝑠𝑛⌋

4

}

∑

𝑗=0

{3

4𝑗} + 𝑏 − 1 2 {4𝑛

𝑐 } −

𝑏 {⌊4𝑠𝑛⌋

𝑏

}

∑

𝑗=0

{−𝑝 𝑏 𝑗} .

Likewise, we may find the Ehrhart quasi-polynomial for|𝑃_{𝑛𝑐𝐷𝑥} ∩ ℤ²|. To simplify our calculations, we may consider𝑛𝑎𝐷_𝑧 = 4𝑛𝐷_𝑧 ∼ 𝑛𝑐𝐷_𝑥. By linear equivalence,|𝑃_𝑛𝑎𝐷

𝑧 ∩ ℤ²| = |𝑃_𝑛𝑐𝐷

𝑥∩ ℤ²|. The polytope of𝑛𝑎𝐷_𝑧is given by 𝑃_{𝑛𝑎𝐷𝑧} = Conv(𝐴^′, 𝐵^′, 𝐶^′) ∶= Conv ((0, 0), (−4𝑝

𝑏 𝑛, −4𝑛) , (3𝑛, −4𝑛)) ,

with𝐴^′𝐶^′contained in the line𝑦 = −⁴

3𝑥and𝐴^′𝐵^′contained in the line𝑦 = ^𝑏

𝑝𝑥. Similarly as before:

|𝑃_𝑛𝑎𝐷

𝑧 ∩ ℤ²| =

∑4𝑛 𝑗=0

(

⌊3 4𝑗

⌋

−⌈ −𝑝 𝑏 𝑗

⌉ + 1)

=

∑4𝑛 𝑗=0

(3 4𝑗 − {3

4𝑗}) −

∑4𝑛 𝑗=0

(−𝑝

𝑏 𝑗 + 1 − {−𝑝 𝑏 𝑗}) +

⌊4𝑛 𝑏

⌋

+ 1 + 4𝑛 + 1

= 𝑐 4𝑏

(4𝑛 + 1 2

)

−3

2𝑛 + 2(𝑏 − 1)

𝑏 𝑛 − 𝑏 − 1 2 {4𝑛

𝑏 } +

𝑏{^4𝑛

𝑏}

∑

𝑗=0

{−𝑝

𝑏 𝑗} + 4𝑛

𝑏 + 1 − {4𝑛 𝑏 } ,

using expressions that we obtained previously. Thus, we can write

|𝑃_{𝑛𝑐𝐷𝑥} ∩ ℤ²| = |𝑃_{𝑛𝑎𝐷𝑧} ∩ ℤ²| = 𝑐₂(𝑐𝐷_𝑥, 𝑛)𝑛²+ 𝑐₁(𝑐𝐷_𝑥, 𝑛)𝑛 + 𝑐₀(𝑐𝐷_𝑥, 𝑛), where

𝑐₂(𝑐𝐷_𝑥, 𝑛) = 2 𝑠, 𝑐₁(𝑐𝐷_𝑥, 𝑛) = 1

2(1 +1 𝑠 + 4

𝑏) 𝑛, 𝑐₀(𝑐𝐷_𝑥, 𝑛) = 1 − {4𝑛

𝑏 } −𝑏 − 1 2 {4𝑛

𝑏 } +

𝑏{^4𝑛

𝑏}

∑

𝑗=0

{−𝑝

𝑏 𝑗} .

Our next goal is to give upper bounds on the constant terms of the Ehrhart quasi-polynomials of|𝑃_{𝑛𝛿𝐷𝑥}∩ ℤ²|,𝛿 = 𝑏, 𝑐. In Proposition 3.1, notice that the expressions of the last two terms of𝑐₀(𝑏𝐷_𝑥, 𝑛) and𝑐₀(𝑐𝐷_𝑥, 𝑛)are of the same form, which we will analyze in depth in Section 4. In the following, we give a uniform upper bound on𝑐₀(𝑏𝐷_𝑥, 𝑛)minus its last two terms.

Lemma 3.2. In the expression of𝑐₀(𝑏𝐷_𝑥, 𝑛), we have

−5 2{𝑠𝑛} +

4 {⌊4𝑠𝑛⌋

4

}

∑

𝑗=0

{3 4𝑗} ≤ 1

8 for all𝑛 ≥ 0, where𝑠 = ^𝑏

𝑐. Furthermore:

(1) The above expression is positive if and only if ¹

4 < {𝑠𝑛} < ³

10. (2) The above expression is greater than ⁻¹

32𝑠 only if{𝑠𝑛} < ¹

2+ ¹

80𝑠. Proof. Let𝑏𝑛 = 𝑚𝑐 + 𝑟with0 ≤ 𝑟 ≤ 𝑐 − 1, so that{𝑠𝑛} = ^𝑟

𝑐. Let𝓁 = 0, 1, 2, 3be the integer such that^𝓁𝑐

4 ≤ 𝑟 < ^(𝓁+1)𝑐

4 . Then⌊4𝑠𝑛⌋ = 4𝑚 + 𝓁and so4 {⌊4𝑠𝑛⌋

4

}

= 𝓁.

Now, we will bound the given expression from the above for each𝓁 = 0, 1, 2, 3. If𝓁 = 0, then the given expression in the lemma is−^5𝑟

2𝑐 ≤ 0. If𝓁 = 1, we have

−^5𝑟

2𝑐 +³

4 ≤ −⁵

2 ⋅ ¹

4 +³

4 = ¹

8. If𝓁 = 2, we have−^5𝑟

2𝑐 +³

4 + ²

4 ≤ −⁵

2 ⋅¹

2 + ⁵

4 = 0. Lastly, if𝓁 = 3, we have−^5𝑟

2𝑐+ ³

4+²

4+ ¹

4 ≤ −⁵

2⋅³

4+ ³

2 = −³

8.

For the final statements of the lemma, we see the expression is non-positive if 𝓁 ≠ 1, and so we must have¹

4 < {𝑠𝑛}. When𝓁 = 1, we computed the expression is equal to−^5𝑟

2𝑐+³

4, which is positive if and only if{𝑠𝑛} = ^𝑟

𝑐 < ³

10.

Similarly, the expression could be greater than_32𝑠⁻¹ in cases𝓁 = 0, 1, 2. (Note that𝑠 < 1because𝑏 < 𝑐.) Working case by case with the expressions obtained, we obtain that{𝑠𝑛} = ^𝑟

𝑐 < ¹

2+ ¹

80𝑠 in order for the expression to be greater than

−1

32𝑠.

Note that−¹

8𝑠({4𝑠𝑛}²− {4𝑠𝑛}) ≤ ¹

32𝑠since the function𝑥 − 𝑥²is maximized at 𝑥 = ¹

2. Combining this observation with Lemma 3.2, we obtain the following corollary.

Corollary 3.3. We have

𝑐₀(𝑏𝐷_𝑥, 𝑛) ≤ 9 8 + 1

32𝑠 + 𝑏 − 1 2 {4𝑛

𝑐 } −

𝑏 {⌊4𝑠𝑛⌋

𝑏

}

∑

𝑗=0

{−𝑝 𝑏 𝑗} . Moreover, if{𝑠𝑛} ≥ ¹

2+ ¹

80𝑠, we may improve the above bound as follows:

𝑐₀(𝑏𝐷_𝑥, 𝑛) ≤ 1 + 𝑏 − 1 2 {4𝑛

𝑐 } −

𝑏 {⌊4𝑠𝑛⌋

𝑏

}

∑

𝑗=0

{−𝑝 𝑏 𝑗} .

4. Bounding𝒄_𝟎(𝒃𝑫_𝒙, 𝒏)and𝒄_𝟎(𝒄𝑫_𝒙, 𝒏)

In this section, we prove the key results needed to bound 𝑐₀(𝑏𝐷_𝑥, 𝑛) and 𝑐₀(𝑐𝐷_𝑥, 𝑛). This amounts to obtaining bounds for the expression ^𝑏−1

2

{4𝑛 𝑐

}

−

∑𝑟 𝑗=0{^−𝑝

𝑏 𝑗}, where𝑟 = 𝑏{^{⌊4𝑠𝑛⌋}

𝑏 }. We begin by recording the following lemma.

Lemma 4.1. Let𝑛, 𝑏, 𝑐 ∈ ℤ⁺with4 < 𝑏 < 𝑐andgcd(4, 𝑏, 𝑐) = 1. Let𝑝 < 0be an integer satisfying4𝑝 + 3𝑏 = 𝑐and𝑠 = ^𝑏

𝑐. If𝑟 = 𝑏 {⌊4𝑠𝑛⌋

𝑏

} , then 𝑏 − 1

2 {4𝑛 𝑐 } −

∑𝑟 𝑗=0

{−𝑝

𝑏 𝑗} ≤ (𝑏 − 1)(𝑟 + 1)

2𝑏 −

∑𝑟 𝑗=0

{−𝑝 𝑏 𝑗} .

Proof. Since𝑟 is the reminder when𝑏 is divided into ⌊4𝑠𝑛⌋, we have4𝑠𝑛 =

⌊4𝑠𝑛⌋ + {4𝑠𝑛} = 𝑏⌊^{⌊4𝑠𝑛⌋}

𝑏 ⌋ + 𝑟 + {4𝑠𝑛}. So, {4𝑛

𝑐 } = {4𝑠𝑛 𝑏 } = {⎢

⎣

⌊4𝑠𝑛⌋

𝑏

⎥

⎦+ 𝑟

𝑏 + {4𝑠𝑛}

𝑏 } = 𝑟 + {4𝑠𝑛}

𝑏 ,

where the last equality uses0 ≤ ^𝑟

𝑏 ≤ ^𝑏−1

𝑏 and0 ≤ ^{4𝑠𝑛}

𝑏 < ¹

𝑏.

By the above lemma, it suffices to bound the expression on the righthand side. In §4.1, we give a general algorithm to obtain bounds on expressions of the form ^{(𝛽−1)(𝑢+1)}

2𝛽 −∑𝑢

𝑗=0{^𝛼

𝛽𝑗} when𝛼and𝛽 satisfy particular Diophantine equations, see Corollary 4.3.

4.1. An algorithm to bound expressions of the form^{(𝜷−𝟏)(𝒖+𝟏)}

𝟐𝜷 −∑𝒖 𝒋=𝟎{^𝜶

𝜷𝒋}.

Our goal in this subsection is to prove:

Proposition 4.2. Suppose that𝛼₀> 𝛼₁,𝛽₀> 𝛽₁, and 𝛼₁𝛽₀− 𝛽₁𝛼₀= 𝜎 = ±1

where𝛼_𝑖, 𝛽_𝑖 ∈ ℤ⁺. Let𝑢₀ = 𝛽₁𝑡₁+ 𝑢₁, where𝑢₀, 𝑢₁, 𝑡₁ ∈ ℤ^≥0and0 ≤ 𝑢_𝑖 < 𝛽_𝑖 for all𝑖. Then

(𝑢₀+ 1)(𝛽₀− 1)

2𝛽₀ −

𝑢₀

∑

𝑗=0

{𝛼₀

𝛽₀𝑗} = (𝑢₁+ 1)(𝛽₁− 1)

2𝛽₁ −

𝑢₁

∑

𝑗=0

{𝛼₁𝑗

𝛽₁ } + 𝜖(𝜎, 𝑡₁, 𝑢₁, 𝛽₀, 𝛽₁), where

𝜖(𝜎, 𝑡, 𝑢, 𝛽^′, 𝛽) = (𝑢 + 1)(𝜎𝑢 + 𝛽^′− 𝛽)

2𝛽^′𝛽 +𝜎𝑡(𝛽(𝑡 − 𝜎) + 2𝑢 + 1 − 𝛽^′) 2𝛽^′

+ ∆(𝜎, 𝑡, 𝑢, 𝛽^′, 𝛽)

and∆(𝜎, 𝑡, 𝑢, 𝛽^′, 𝛽) = 1if𝛽^′− 𝛽 ≤ 𝛽𝑡 + 𝑢and𝜎 = −1, and∆(𝜎, 𝑢, 𝛽^′, 𝛽) = 0 otherwise.

When applied iteratively, we arrive at the following algorithm.

Corollary 4.3. Suppose we have sequences of positive integers𝛼₀ > 𝛼₁ > ⋯ >

𝛼_𝑁and𝛽₀> 𝛽₁ > ⋯ > 𝛽_𝑁such that for all𝑖,

𝛼_𝑖𝛽_𝑖−1− 𝛽_𝑖𝛼_𝑖−1= 𝜎_𝑖 = ±1.

Let𝑢₀, … , 𝑢_𝑁and𝑡₁, … , 𝑡_𝑁be non-negative integers satisfying𝑢_𝑖−1 = 𝛽_𝑖𝑡_𝑖+𝑢_𝑖and 0 ≤ 𝑢_𝑖 < 𝛽_𝑖for all𝑖. Then

(𝑢₀+ 1)(𝛽₀− 1)

2𝛽₀ −

𝑢₀

∑

𝑗=0

{𝛼₀

𝛽₀𝑗} = (𝑢_𝑁 + 1)(𝛽_𝑁− 1) 2𝛽_𝑁

−

𝑢_𝑁

∑

𝑗=0

{𝛼_𝑁𝑗 𝛽_𝑁 } +

∑𝑁 𝑖=1

𝜖(𝜎_𝑖, 𝑡_𝑖, 𝑢_𝑖, 𝛽_𝑖−1, 𝛽_𝑖), where𝜖is as in Proposition4.2.

We begin with the following preliminary lemmas.

Lemma 4.4. Let𝛼, 𝛽 ∈ ℤ⁺be relatively prime. Then

𝛽−1∑

𝑘=0

⎢⎢

⎣ 𝛼 𝛽𝑘

⎥⎥

⎦

= 1

2(𝛼 − 1)(𝛽 − 1) and

𝛽−1∑

𝑘=0

⎡

⎢⎢ 𝛼 𝛽𝑘⎤

⎥⎥

= 1

2(𝛼 + 1)(𝛽 − 1).

Proof. Notice that ∑𝛽 𝑘=0⌊^𝛼

𝛽𝑘⌋ + 𝛽 + 1is the number of lattice points in the triangle with vertices0,(𝛽, 0), and(𝛽, 𝛼). So, by Pick’s Theorem,

∑𝛽 𝑘=0

⎢⎢

⎣ 𝛼 𝛽𝑘⎥

⎥

⎦

+ 𝛽 + 1 = 1

2(𝛼𝛽 + 𝛼 + 𝛽 + 1) + 1.

Since∑𝛽 𝑘=0⌊^𝛼

𝛽𝑘⌋ = ∑𝛽−1 𝑘=0⌊^𝛼

𝛽𝑘⌋ + 𝛼, the first result follows. The second result follows from the first and the fact that∑𝛽−1

𝑘=0⌈^𝛼

𝛽𝑘⌉ = (𝛽 − 1) +∑𝛽−1 𝑘=0⌊^𝛼

𝛽𝑘⌋.

Lemma 4.5. Suppose that𝛼₀ > 𝛼₁,𝛽₀ > 𝛽₁, and 𝛼₁𝛽₀− 𝛽₁𝛼₀= 𝜎 = ±1 where𝛼_𝑖, 𝛽_𝑖 ∈ ℤ⁺. Then

(1) {^𝛼0𝑗

𝛽₀} = {^𝛼1𝑗

𝛽₁ } − 𝜎 ^𝑗

𝛽₁𝛽₀ for all0 ≤ 𝑗 < 𝛽₁, and (2) for any integer𝑢satisfying0 ≤ 𝑢 < 𝛽₁,

∑𝑢 𝑗=0

{𝛼₀𝑗 𝛽₀ } =

∑𝑢 𝑗=0

{𝛼₁𝑗 𝛽₁ } − 𝜎

𝛽₀𝛽₁ (𝑢 + 1

2 )

(3)

𝛽₁

∑

𝑗=0

{𝛼₀𝑗

𝛽₀ } = 1 − 𝜎 + (𝛽₁+ 𝜎)(𝛽₀− 𝜎)

2𝛽₀ .

Proof. We begin with the proof of (1). The case𝑗 = 0is clear, so we assume 1 ≤ 𝑗 ≤ 𝛽₁− 1. Since^𝛼0𝑗

𝛽₀ = ^𝛼1𝑗

𝛽₁ − 𝜎 ^𝑗

𝛽₁𝛽₀, it suffices to show0 ≤ {^𝛼1𝑗

𝛽₁ } − 𝜎 ^𝑗

𝛽₁𝛽₀ < 1. Since𝛼₁and𝛽₁are relatively prime, we see ¹

𝛽₁ ≤ {^𝛼1𝑗

𝛽₁ } ≤ 1 − ¹

𝛽₁. The result then follows from the fact that|𝜎 ^𝑗

𝛽₁𝛽₀| < ¹

𝛽₀ < ¹

𝛽₁. Part (2) follows from (1) by noting

∑𝑢 𝑗=0

{𝛼₀𝑗 𝛽₀ } =

∑𝑢 𝑗=0

({𝛼₁𝑗

𝛽₁ } − 𝜎 𝑗 𝛽₀𝛽₁) =

∑𝑢 𝑗=0

{𝛼₁𝑗 𝛽₁ } − 𝜎

𝛽₀𝛽₁ (𝑢 + 1

2 )

.

To prove (3), we use∑𝛽₁ 𝑗=0{^𝛼0𝑗

𝛽₀ } = ^𝜎+1

2 − ^𝜎

𝛽₀ +∑𝛽₁−1 𝑗=0 {^𝛼0𝑗

𝛽₀ }and part (2) to see