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Dynamics of generalized Farey sequences with applications to equidistribution

广义 Farey 序列的动力学及其在均匀分布中的应用

基本信息

批准号：
EP/T005130/1
负责人：
Jimmy Tseng
金额：
$ 15.92万
依托单位：
UNIVERSITY OF EXETER
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT005130%2F1
关键词：
Dynamics generalized Farey sequences applications

项目摘要

The Farey sequence, discovered in the early nineteenth century, is now an important object in number theory, geometry, and homogeneous dynamics. For example, a conjecture concerning the distribution of Farey sequences is equivalent to the Riemann hypothesis, one of the most famous outstanding problems in number theory and, indeed, all of mathematics. It also plays a key role in approximation in number theory through its connections with objects called continued fractions and an important role in geometry and homogenous dynamics through its connections with the dynamics of horocycles, which are important curves in hyperbolic geometry. This leads to an elegant theory in number theory, geometry, and homogeneous dynamics whose link is the Farey sequence, whose applications are found in, for example, mathematical physics and applied dynamics, and whose influence reach out towards physics and biology. The Farey sequence is easy to describe. Let n be a natural number. The Farey sequence of order n is a sequence of rational numbers in lowest terms between 0 and 1 with denominators less than or equal to n, ordered by increasing size. For example, the Farey sequence of order 1 is the sequence {0, 1} and the Farey sequence of order 2 is the sequence {0, 1/2, 1}. These sequences give rise, via hyperbolic geometry, to continued fractions which yield the best approximations of a given real number. Such approximations are a central concern in the subfield of number theory called Diophantine approximation. These sequences also are important in hyperbolic geometry itself because they help us understand horocycles and, in particular, how horocycles distribute under the dynamics of a natural flow, namely the geodesic flow. In this way, Farey sequences provide a deep link between number theory, homogeneous dynamics, and geometry, a link which should be generalised and deepened further to the benefit of all three fields.This proposal aims to generalise and deepen this link, and the expected results will belong to three fields, multiplying their benefit. We will study the analog of the Farey sequence in very general settings such as spaces coming from locally compact Hausdorff (topological) groups and their discrete subgroups. These can be large and complicated spaces. Topological groups are spaces for which we have a notion of multiplication, namely any two elements multiplied together yields another element in the group, and which satisfy some sensible rules. Locally compact and Hausdorff are two topological notions and many interesting spaces, such as the plane, three-dimensional space or, more generally, manifolds, have these properties. We will then use these generalized Farey sequences in two ways. The first is to study their number-theoretic properties in analogy with the classical Farey sequences and the rich number theory coming from them. The second is to use these sequences to study dynamics on these large and complicated spaces again in analogy with the classical Farey sequences and horocycles. Some of the tools that we will use are the mixing property coming from ergodic theory, Ratner's theorems coming from homogenous dynamics, Eisenstein series coming from analytic number theory, and harmonic analysis, which is a field of mathematics concerned with decomposing functions into the infinite sum of "wave-like functions.'' By generalising the elegant theory of which the Farey sequence is the link, we will also expand upon the applications and influences of the theory.

Farey数列发现于19世纪初，现在是数论、几何学和齐次动力学中的一个重要对象。例如，一个关于Farey序列分布的猜想等同于黎曼假设，这是数论中最著名的突出问题之一，实际上，也是所有数学中最著名的问题之一。它还通过它与被称为连分式的对象的联系在数论的逼近中发挥关键作用，并通过它与双曲几何中重要的圆周动力学的联系在几何和齐次动力学中发挥重要作用。这导致了数论、几何学和齐次动力学中一种优雅的理论，其联系是Farey序列，其应用可以在例如数学物理和应用动力学中找到，其影响延伸到物理和生物学。Farey序列很容易描述。设n是自然数。N阶Farey序列是分母小于或等于n的最小项为0到1的有理数序列，按大小递增排序。例如，1阶Farey序列是序列{0，1}，2阶Farey序列是序列{0，1/2，1}。这些序列通过双曲几何产生产生给定实数的最佳逼近的连分式。这种近似是数论中称为丢番图近似的子域中的一个中心问题。这些序列在双曲几何中也很重要，因为它们帮助我们了解周期，特别是在自然流(即测地线流)的动态下，周期如何分布。通过这种方式，Farey序列在数论、齐次动力学和几何之间提供了一种深入的联系，这种联系应该被进一步推广和深化，以使所有这三个领域都受益。本提案旨在概括和深化这种联系，预期的结果将属于三个领域，乘以它们的好处。我们将在非常一般的环境中研究Farey序列的模拟，例如来自局部紧Hausdorff(拓扑)群及其离散子群的空间。这些空间可能很大，也可能很复杂。拓扑群是我们对其有乘法概念的空间，即任何两个元素相乘产生群中的另一个元素，并且满足某些合理的规则。局部紧和Hausdorff是两个拓扑概念，许多有趣的空间，如平面、三维空间或更一般的流形，都具有这些性质。然后，我们将以两种方式使用这些广义Farey序列。一是类比经典Farey序列及其丰富的数论，研究它们的数论性质。第二种是利用这些序列来研究这些大而复杂的空间上的动力学，类似于经典的Farey序列和周期。我们将使用的一些工具是来自遍历理论的混合性质，来自齐次动力学的Ratner定理，来自解析数论的Eisenstein级数，以及调和分析，调和分析是一个涉及将函数分解为“波函数”的无限和的数学领域。通过推广以Farey序列为纽带的优雅理论，我们还将扩展该理论的应用和影响。