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A weighted vertical Sato-Tate law for Maaß forms on $rm{GSp}_4$ $rm{GSp}_4$上Maaß形式的加权垂直萨托-塔特定律
Pub Date : 2024-09-09 DOI: arxiv-2409.06027
Félicien Comtat
We prove a weighted Sato-Tate law for the Satake parameters of automorphicforms on $rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$whose level tends to infinity. When the level is squarefree we refine ourresult to the cuspidal spectrum. The ingredients are the $rm{GSp}_4$ Kuznetsovformula and the explicit calculation of local integrals involved in theWhittaker coefficients of $rm{GSp}_4$ Eisenstein series. We also discuss howthe problem of bounding the continuous spectrum in the level aspect naturallyleads to some combinatorial questions involving the double cosets in $Pbackslash G / H$, for each parabolic subgroup $P$.
我们证明了$rm{GSp}_4$上自形形的加权萨托-塔特定律,该定律是关于水平趋于无穷大的一般同余子群$H$的。当水平无平方时,我们将结果细化为尖顶谱。其要素是$rm{GSp}_4$库兹涅佐夫公式和$rm{GSp}_4$爱森斯坦级数的维特克系数所涉及的局部积分的显式计算。我们还讨论了在水平方面约束连续谱的问题如何自然地引出一些组合问题,这些问题涉及每个抛物线子群 $P$ 的 $Pbackslash G / H$ 中的双余弦。
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引用次数: 0
Low Discrepancy Digital Kronecker-Van der Corput Sequences 低差分数字克罗内克-范德科普特序列
Pub Date : 2024-09-09 DOI: arxiv-2409.05469
Steven Robertson
The discrepancy of a sequence measures how quickly it approaches a uniformdistribution. Given a natural number $d$, any collection of one-dimensionalso-called low discrepancy sequences $left{S_i:1le i le dright}$ can beconcatenated to create a $d$-dimensional $textit{hybrid sequence}$$(S_1,dots,S_d)$. Since their introduction by Spanier in 1995, manyconnections between the discrepancy of a hybrid sequence and the discrepancy ofits component sequences have been discovered. However, a proof that a hybridsequence is capable of being low discrepancy has remained elusive. This paperremedies this by providing an explicit connection between Diophantineapproximation over function fields and two dimensional low discrepancy hybridsequences. Specifically, let $mathbb{F}_q$ be the finite field of cardinality $q$. Itis shown that some real numbered hybrid sequence$mathbf{H}(Theta(t),P(t)):=textbf{H}(Theta,P)$ built from the digitalKronecker sequence associated to a Laurent series$Theta(t)inmathbb{F}_q((t^{-1}))$ and the digital Van der Corput sequenceassociated to an irreducible polynomial $P(t)inmathbb{F}_q[t]$ meets theabove property. More precisely, if $Theta(t)$ is a counterexample to the socalled $t$$textit{-adic Littlewood Conjecture}$ ($t$-$LC$), then anotherLaurent series $Phi(t)inmathbb{F}_q((t^{-1}))$ induced from $Theta(t)$ and$P(t)$ can be constructed so that $mathbf{H}(Phi,P)$ is low discrepancy. Suchcounterexamples to $t$-$LC$ are known over a number of finite fields by, on theone hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and theauthor.
序列的差异度量它接近均匀分布的速度。给定一个自然数 $d$,任何一维的所谓低差异序列 $left{S_i:1le ile dright}$ 的集合都可以合并成一个 $d$ 维的 $textit{hybrid sequence}$$(S_1,dots,S_d)$。自从斯潘尼尔在 1995 年提出混合序列以来,人们发现了混合序列的差异与其组成序列的差异之间的许多联系。然而,混合序列能够低差异的证明却一直没有找到。本文通过提供函数域上的 Diophantineapproximation 与二维低差异混合序列之间的明确联系来解决这个问题。具体地说,让 $mathbb{F}_q$ 是心数为 $q$ 的有限域。结果表明,某个实数混合序列$mathbf{H}(Theta(t),P(t)):=textbf{H}(Theta,P)$ 由与劳伦数列$Theta(t)inmathbb{F}_q((t^{-1}))$相关联的数字克朗内克尔序列和与不可约多项式$P(t)inmathbb{F}_q[t]$相关联的数字范德尔科普特序列建立,满足上述性质。更确切地说,如果 $Theta(t)$ 是所谓的 $t$$textit{-adic Littlewood Conjecture}$ ($t$$-$LC$)的反例、那么由 $Theta(t)$ 和 $P(t)$ 引起的另一个洛伦级数 $Phi(t)inmathbb{F}_q((t^{-1}))$ 可以被构造出来,从而使 $mathbf{H}(Phi,P)$ 是低差异的。在许多有限域上,这种 $t$-$LC$ 的反例是已知的,一方面是阿迪森、内沙林和卢农,另一方面是加勒特和作者。
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引用次数: 0
Uniform Discreteness of Discrete Orbits of Non-Uniform Lattices in $SL_2(mathbb{R})$ $SL_2(mathbb{R})$中非均匀网格的离散轨道的均匀不严密性
Pub Date : 2024-09-09 DOI: arxiv-2409.05935
Sahar Bashan
We study the property of uniform discreteness within discrete orbits ofnon-uniform lattices in $SL_2(mathbb{R})$, acting on $mathbb{R}^2$ by lineartransformations. We provide a new proof of the conditions under which the orbitof a non-uniform lattice in $SL_2(mathbb{R})$ is uniformly discrete, by usingDiophantine properties. Our results include a detailed analysis of theasymptotic behavior of the error terms. Focusing on a specific group $Gamma$and a discrete orbit of it, $S$, the main result of this paper is that for any$epsilon>0$, three points in $S$ can be found on a horizontal line withindistance $epsilon$ of each other. This gives a partial result toward aconjecture of Leli`evre. The set $S$ and group $Gamma$ are respectively theset of long cylinder holonomy vectors, and Veech group, of the "golden L"translation surface.
我们研究了通过线性变换作用于 $mathbb{R}^2$ 的 $SL_2(mathbb{R})$ 中的非均匀网格的离散轨道内均匀离散性的性质。我们利用二凡性质,对$SL_2(mathbb{R})$中的非均匀网格的轨道均匀离散的条件进行了新的证明。我们的结果包括对误差项渐近行为的详细分析。本文聚焦于一个特定的组 $Gamma$ 及其离散轨道 $S$,其主要结果是:对于任意 $epsilon>0$,可以在水平线上找到 $S$中的三个点,它们之间的距离为 $epsilon$。这给出了勒利耶夫猜想的部分结果。集合$S$和群$Gamma$分别是 "黄金L "平移面的长圆柱整体矢量集合和Veech群。
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引用次数: 0
Ergodicity and Algebraticity of the Fast and Slow Triangle Maps 快慢三角映射的遍历性和代数性
Pub Date : 2024-09-09 DOI: arxiv-2409.05822
Thomas Garrity, Jacob Lehmann Duke
Our goal is to show that both the fast and slow versions of the triangle map(a type of multi-dimensional continued fraction algorithm) in dimension $n$ areergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. Thisparticular type of higher dimensional multi-dimensional continued fractionalgorithm has recently been linked to the study of partition numbers, with theresult that the underlying dynamics has combinatorial implications.
我们的目标是证明三角形映射(一种多维连续分数算法)在 $n$ 维度下的快速和慢速版本都是连续的,从而解决 Messaoudi、Noguiera 和 Schweiger 的一个猜想。这种特殊类型的高维多维连续分数算法最近与分割数的研究联系在一起,其结果是基本动力学具有组合意义。
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引用次数: 0
Asymptotics for smooth numbers in short intervals 短区间平稳数的渐近线
Pub Date : 2024-09-09 DOI: arxiv-2409.05761
Khalid Younis
A number is said to be $y$-smooth if all of its prime factors are less thanor equal to $y.$ For all $17/30
如果一个数的所有质因数都小于或等于 $y,那么这个数就被称为 $y$光滑数。对于所有的$17/30
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引用次数: 0
A Database of Continued Fractions of Polynomial Type 多项式类型的连续分数数据库
Pub Date : 2024-09-09 DOI: arxiv-2409.06086
Henri Cohen
We describe a database of 1307 continued fractions with polynomialcoefficients, of which more than 1000 are new, both for interesting constantsand for transcendental functions, and provide the database inside the TeXsource of the paper. Look in particular at the section ``Table of Contents''.
我们描述了一个包含 1307 个多项式系数的续分数数据库,其中超过 1000 个是新的,既有有趣的常数,也有超越函数,并在论文的 TeX 源中提供了该数据库。请特别查看 "目录 "部分。
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引用次数: 0
Algorithms for complementary sequences 互补序列算法
Pub Date : 2024-09-09 DOI: arxiv-2409.05844
Chai Wah Wu
Finding the n-th positive square number is easy, as it is simply $n^2$. Buthow do we find the complementary sequence, i.e. the n-th positive nonsquarenumber? For this case there is an explicit formula. However, for generalconstraints on a number, this is harder to find. In this brief note, we studyhow to compute the n-th integer that does (or does not) satisfy a certaincondition. In particular, we consider it as a fixed point problem, relate it tothe iterative method of Lambek and Moser, study a bisection approach to thisproblem and provide a new formula for the n-th non-k-th power.
找到 n 次正平方数很容易,因为它就是 $n^2$。但如何求互补序列,即 n 次正非平方数呢?对于这种情况,有一个明确的公式。然而,对于一个数的一般约束条件,这就比较难找了。在这篇短文中,我们将研究如何计算满足(或不满足)某个条件的第 n 个整数。特别是,我们将其视为一个定点问题,将其与 Lambek 和 Moser 的迭代法联系起来,研究了解决这一问题的分段方法,并提供了一个新的非 k 次幂的 n 次整数公式。
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引用次数: 0
Large degree primitive points on curves 曲线上的大度原始点
Pub Date : 2024-09-09 DOI: arxiv-2409.05796
Maarten Derickx
A number field $K$ is called primitive if $mathbb Q$ and $K$ are the onlysubfields of $K$. Let $X$ be a nice curve over $mathbb Q$ of genus $g$. Apoint $P$ of degree $d$ on $X$ is called primitive if the field of definition$mathbb Q(P)$ of the point is primitive. In this short note we prove that if$X$ has a divisor of degree $d> 2g$, then $X$ has infinitely many primitivepoints of degree $d$. This complements the results of Khawaja and Siksek thatshow that points of low degree are not primitive under certain conditions.
如果 $mathbb Q$ 和 $K$ 是 $K$ 的唯一子域,则数域 $K$ 称为原始域。让 $X$ 是一条在 $mathbb Q$ 上的好曲线,其属为 $g$。如果点的定义域$mathbb Q(P)$是原始的,那么在$X$上度为$d$的点$P$就称为原始点。在这篇短文中,我们证明如果$X$有一个度数为$d> 2g$的除数,那么$X$就有无限多个度数为$d$的原始点。这是对哈瓦贾和西克塞克的结果的补充,哈瓦贾和西克塞克的结果表明,在某些条件下,低度点不是原始点。
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引用次数: 0
Bounded distance equivalence of cut-and-project sets and equidecomposability 切分与投影集合的有界距离等价性和可等价分解性
Pub Date : 2024-09-09 DOI: arxiv-2409.05450
Sigrid Grepstad
We show that given a lattice $Gamma subset mathbb{R}^m timesmathbb{R}^n$, and projections $p_1$ and $p_2$ onto $mathbb{R}^m$ and$mathbb{R}^n$ respectively, cut-and-project sets obtained using Jordanmeasurable windows $W$ and $W'$ in $mathbb{R}^n$ of equal measure are boundeddistance equivalent only if $W$ and $W'$ are equidecomposable by translationsin $p_2(Gamma)$. As a consequence, we obtain an explicit description of thebounded distance equivalence classes in the hulls of simple quasicrystals.
我们证明,给定一个网格 $Gamma 子集 mathbb{R}^m timesmathbb{R}^n$, 以及投影 $p_1$ 和 $p_2$ 分别到 $mathbb{R}^m$ 和 $mathbb{R}^n$ 上、只有当$W$和$W'$通过在$p_2(Gamma)$中平移可等价分解时,使用等度量的$W$和$W'$在$mathbb{R}^n$中得到的乔丹可度量窗口$W$和$W'$的切分与投影集合才是有界距离等价的。因此,我们得到了简单准晶体壳中有界距离等价类的明确描述。
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引用次数: 0
Zeta functions enumerating subforms of quadratic forms 枚举二次函数子形式的 Zeta 函数
Pub Date : 2024-09-09 DOI: arxiv-2409.05625
Daejun Kim, Seok Hyeong Lee, Seungjai Lee
In this paper, we introduce and study the Dirichlet series enumerating(proper) equivalence classes of full rank subforms/sublattices of a givenquadratic form/lattice, focusing on the positive definite binary case. Weobtain formulas linking this Dirichlet series with Dirichlet series countingideal classes of the imaginary quadratic field associated with the quadraticform. Utilizing the result, we provide explicit formulas of the Dirichletseries for several lattices, including square lattice and hexagonal lattice.Moreover, we investigate some analytic properties of this Dirichlet series.
在本文中,我们介绍并研究了枚举给定二次型/格的满级子型/子格的(适当)等价类的狄利克特数列,重点是正定二元情况。我们获得了将该狄利克特数列与计算与二次型相关的虚二次型场的理想类的狄利克特数列联系起来的公式。利用这一结果,我们提供了几种晶格(包括正方形晶格和六角形晶格)的 Dirichlet 列的明确公式。此外,我们还研究了该 Dirichlet 列的一些解析性质。
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引用次数: 0
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arXiv - MATH - Number Theory
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