Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters Francesco CellarosiTariq Osman.

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Heavy tailed and compactly supported distributions

of quadratic Weyl sums with rational parameters

Francesco Cellarosi∗† Tariq Osman‡.

January 27, 2023

Abstract

We consider quadratic Weyl sums SN(x;α, β) = PN

n=1 exp2πi 1

2n2+βnx+αn

for (α, β)∈Q2, where x∈Ris randomly distributed according to a probability measure

absolutely continuous with respect to the Lebesgue measure. We prove that the limiting

distribution in the complex plane of 1

√NSN(x;α, β) as N→ ∞ is either heavy tailed or

compactly supported, depending solely on α, β. In the heavy tailed case, the probability

(according to the limiting distribution) of landing outside a ball of radius Ris shown

to be asymptotic to T(α, β)R−4, where the constant T(α, β)>0 is explicit. The result

follows from an analogous statement for products of generalized quadratic Weyl sums of

the form Sf

N(x;α, β) = Pn∈Zfn

Nexp2πi 1

2n2+βnx+αnwhere fis regular. The

precise tails of the limiting distribution of 1

NSf1

NSf2

N(x;α, β) as N→ ∞ can be described

in terms of a measure –which depends on (α, β)– of a super level set of a product of two

Jacobi theta functions on a noncompact homogenous space. Such measures are obtained

by means of an equidistribution theorem for rational horocycle lifts to a torus bundle over

the unit tangent bundle to a cover of the classical modular surface. The cardinality and

the geometry of orbits of rational points of the torus under the aﬃne action of the theta

group play a crucial role in the computation of T(α, β). This work complements and

extends [6] and [32], in which the cases (α, β)/∈Q2and α=β= 0 are considered.

Contents

1 Introduction 2

2 Preliminaries 9

2.1 The Heisenberg Group and the Schr¨odinger Representation . . . . . . . . . . . . 9

2.2 f

SL(2,R) and the Shale-Weil Representation . . . . . . . . . . . . . . . . . . . . 9

2.3 From the Universal Jacobi Group to the Theta Function . . . . . . . . . . . . . 11

∗Department of Mathematics and Statistics. Queen’s University. Kingston, ON, Canada.

†Corresponding author: fc19@queensu.ca

‡Department of Mathematics. Brandeis University. Waltham, MA, U.S.A.

arXiv:2210.09838v3 [math.NT] 25 Jan 2023

2.4 InvarianceProperties................................. 14

2.5 FundamentalDomains ................................ 15

2.6 Geodesic and Horocycle Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Thekeyidentity.................................... 16

3 Invariant Measures 17

3.1 Five prototypical examples of orbits . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 The Normalising Constants |S(α,β)|......................... 22

3.3 Symmetry....................................... 32

4 Limit Theorems 32

4.1 Rational Limit Theorems for Regular Indicators . . . . . . . . . . . . . . . . . . 32

4.2 Rational Limit Theorems for Sharp Indicators . . . . . . . . . . . . . . . . . . . 37

5 Growth in the Cusps and Tail Asymptotics 38

5.1 Tail asymptotics at i∞................................ 40

5.2 Tailasymptoticsat1................................. 43

5.3 Combined tail asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 The main constant 2|U(α,β)|+|V(α,β)|

|S(α,β)|47

6.1 The constant |U(α,β)|................................. 48

6.2 The constant |V(α,β)|................................. 49

6.3 The computation of the leading constant . . . . . . . . . . . . . . . . . . . . . . 51

7 The main theorems 53

7.1 The tails of the limiting distribution for regular indicators . . . . . . . . . . . . 53

7.2 The tails of the limiting distribution for sharp indicators . . . . . . . . . . . . . 54

8 Some numerical illustrations 58

References 61

1 Introduction

We consider quadratic Weyl sums of the form

SN(x;α, β) =

n=1

e1

2n2+βnx+αn,(1.1)

where e(z) = e2πiz,Nis a positive integer, and x,αand βare real. In our analysis, we ﬁx α

and β, and we assume that xin (1.1) is randomly distributed on Raccording to a probability

measure λ, absolutely continuous with respect to the Lebesgue measure. We can understand

sums (1.1) as the position after Nsteps of a deterministic walk in Cwith a random seed x.

-4-2 0 2

-8

-6

-4

-2

α=1/5, β=1/3,

N=100, x=0.874663

-6-4-2 0 2 4 6

-8

-6

-4

-2

α=1/5, β=1/3,

N=100, x=0.317401

-2-101234

-6

-4

-2

α=1/5, β=1/3,

N=100, x=0.0418017

-8-6-4-2 0

-8

-6

-4

-2

α=1/5, β=1/3,

N=100, x=0.506402

-15 -10 -5 0

-2

α=1/2, β=3 ,

N=100, x=0.834469

0 2 4 6 8 10

α=1/2, β=3 ,

N=100, x=0.789321

-2-101234

-10

-5

α=1/2, β=3 ,

N=100, x=0.00687084

-20246

-2

α=1/2, β=3 ,

N=100, x=0.460187

-5-4-3-2-1 0 1

-6

-4

-2

α=1/6, β=1/6,

N=100, x=0.575034

-5 0 5 10 15

-10

-5

α=1/6, β=1/6,

N=100, x=0.332542

-15 -10 -5 0

-2

-1

α=1/6, β=1/6,

N=100, x=0.42748

-3-2-1 0 1 2 3 4

α=1/6, β=1/6,

N=100, x=0.0230083

-12 -10 -8-6-4-2 0

α=1/8, β=0,

N=100, x=0.032655

0 2 4 6 8

-2

α=1/8, β=0,

N=100, x=0.37372

-10 -8-6-4-2 0

-10

-5

α=1/8, β=0,

N=100, x=0.628386

-2 0 2 4 6

-2

α=1/8, β=0,

N=100, x=0.24871

Figure 1: Each panel represents the partial sums of (1.1), i.e. S1(x;α, β), S2(x;α, β), . . . , SN(x;α, β),

where N= 100 and α, β, x are speciﬁed in each panel. Each instance of xhas been randomly sampled

from the uniform measure on [0,1].

Each step is of length 1, and the n-th step is in the direction of e1

2n2+βnx+αn, see

Figure 1.

The emergence of spiral-like structures (curlicues) at various scales in the curves generated

by interpolating the partial sums (1.1) is well understood and can be explained by means

of an approximate functional equation. See, e.g., the works of Hardy–Littlewood [23] [22],

Mordell [37], Wilton [44], Fiedler–Jurkat–K¨orner [17], Deshouillers [10], Coutsias–Kazarinoﬀ

[7] [8], Berry–Goldberg [1], Fedotov–Klopp [16], and Sinai [41]. We investigate the distribution

in the complex plane of the rescaled sums 1

√NSN(x;α, β), as N→ ∞. Analyses along these

lines have been done already in several cases. Jurkat–van Horne [27] [28] [26] studied the

limiting distribution (in R) of 1

√N|SN(x; 0,0)|when xis uniformly distributed on an interval

and found asymptotic formulas for all the moments (with explicit dependence on the interval)

as N→ ∞. The limiting distribution (in C) of 1

√NSN(x; 0,0) was studied by Marklof [31]

[32] for general sampling measures λ. In [31], he also discussed the case β= 0 and found

that the limiting distribution depends on whether αis rational or irrational. The existence of

the limiting distributions (in Ck) of 1

√NSbt1Nc(x; 0,0), Sbt2Nc(x; 0,0), . . . , SbtkNc(x; 0,0), where

0≤t1≤t2≤tk, was proven by Cellarosi [4]. An explicit limiting stochastic process for

t7→ 1

√NSbtNc(x;α, β) was found by Cellarosi–Marklof [6] when (α, β)/∈Q2. In this case, it

was proven in [6] that the limiting distribution of 1

√NSN(x;α, β) is heavy tailed. Speciﬁcally, if

(α, β)/∈Q2, we have

lim

N→∞ λnx∈R:1

√N|SN(x;α, β)|> Ro=6

π2R61 + O(R−12

31 )(1.2)

as R→ ∞. For comparison, when α=β= 0, it was shown in [32] that

lim

N→∞ λnx∈R:1

√N|SN(x; 0,0)|> Ro=4 log 2

π2R4(1 + o(1)) (1.3)

as R→ ∞. Both (1.2)-(1.3) were recently sharpened by Cellarosi–Griﬃn–Osman [5] by replac-

ing the terms in parentheses by (1 + Oε(R−2+ε)) for every ε > 0. For the limiting behaviour of

SN(x;α, β) for ﬁxed xand random (α, β), see the work of Griﬃn–Marklof [20].

In this work we complement (1.2)-(1.3), as well as the results of [5], by considering arbitrary

(α, β)∈Q2. We focus on the leading behaviour of the tails of the limiting distribution and its

explicit dependence on (α, β). Before stating our results, let us introduce some notation.

Deﬁnition 1.0.1. Denote by {u} ∈ [0,1) the fractional part of a real number u. Given

(α, β)∈Q2, the numbers a, b, q ∈Nsuch that {α}=a

q,{β}=b

q, and gcd(a, b, q) = 1 are

unique. We shall call qthe denominator of (α, β) and (a, b)∈ {0,1, . . . , q −1}2the numerators

of (α, β). Moreover, we shall refer to rationals pairs whose denominator is congruent to 2 modulo

4 and whose numerators are both odd as type C. In this case we shall write (α, β)∈C(2m) to

indicate that (α, β) has denominator 2mand is of type C(here mis necessarily odd). We shall

refer too all rational pairs that are not of type Cas type H. We shall write (α, β)∈H(q) to

indicate that the pair (α, β) has denominator qand is of type H.

Let us also introduce the Dedekind ψ-function

ψ(n) = nY

p|n1 + 1

p.(1.4)

We prove the following

Theorem 1.0.2. Let (α, β)∈Q2. Suppose λis a Borel probability measure on Rabsolutely

continuous with respect to Lebesgue measure. Let r≥1.

(i) If (α, β)∈C(2m), then there exists R0=R0(m, r)>0such that if R > R0then we have

lim

N→∞ λx∈R:1

NSNSbrN c(x;α, β)> R2= 0.(1.5)

In other words, the limiting distribution of 1

NSNSbrN c(·;α, β)is compactly supported.

(ii) If (α, β)∈H(q), then there exists T(q;r)>0such that

lim

N→∞ λx∈R:1

NSNSbrN c(x;α, β)> R2∼ T (q;r)R−4,(1.6)

as R→ ∞. In particular, the limiting distribution of |1

NSNSbrN c(·;α, β)|has heavy tails.

Moreover, we have

T(q;r) = C(q)Drat(r)

π2,(1.7)

where, writing q= 2`mwith `≥0and modd,

C(q) = 









ψ(m)if `= 0 or (`= 1 and either aor bis even);

2`−1ψ(m)if ` > 1,

(1.8)

ψis as in (1.4), and

Drat(r) = (2 log 2 if r= 1;

2rcoth−1(r) + 1

2log(r2−1) + r2

2log(1 −1

r2)if r > 1.(1.9)

Example 1.0.3. The rational pairs (1

2,1

2),(1

6,1

6),(1

2,1

6) = (3

6,1

6),(5

6,1

2),(1

10 ,1

6) = ( 3

30 ,5

30 ) are all

of type C. Note that all integer pairs have denominator 1 and hence Z2=H(1). We have

C(1) = 2. A few more examples:

•(0,1

2),(1

2,0),(−3

2,0) ∈H(2) and C(2) = 2,

•(a

5,b

5)∈H(5) for all (a, b)∈Z2r(5Z)2and C(5) = 1

•(0,1

6),(1

3,1

6)=(2

6,1

6),(1

2,1

3)=(3

6,2

6)∈H(6) and C(6) = 1

•(3

2,4

5) = (15

10 ,8

10 )∈H(10) and C(10) = 1

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HeavytailedandcompactlysupporteddistributionsofquadraticWeylsumswithrationalparametersFrancescoCellarosi*TariqOsman.January27,2023AbstractWeconsiderquadraticWeylsumsSN(x;;)=PNn=1exp2i12n2+nx+nfor(;)2Q2,wherex2Risrandomlydistributedaccordingtoaprobabilitymeasureabsolutelycontinuouswithre...

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Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters Francesco CellarosiTariq Osman.

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