Dynamics of generalized Farey sequences with applications to equidistribution

在世纪早期发现的法里数列，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序列分布的猜想等价于黎曼假设，这是数论中最著名的突出问题之一，事实上，是所有数学中最著名的突出问题之一。它还通过与称为连分数的对象的联系在数论中的近似中发挥了关键作用，并通过与双曲几何中重要曲线的horcycle动力学的联系在几何和齐次动力学中发挥了重要作用。这导致了一个优雅的理论在数论，几何，和齐次动力学的联系是法雷序列，其应用中发现，例如，数学物理和应用动力学，其影响力达到物理学和生物学。法里序列很容易描述。设n为自然数。n阶法瑞数列是一个有理数数列，其最低项在0和1之间，且其乘数小于或等于n，按大小递增排序。例如，1阶的Farey序列是序列{0，1}，2阶的Farey序列是序列{0，1/2，1}。这些序列产生，通过双曲几何，连续分数产生最佳的近似一个给定的真实的号码。这种近似是数论中称为丢番图近似的子领域的核心问题。这些序列在双曲几何中也很重要，因为它们帮助我们理解了单圈，特别是在自然流（即测地线流）的动力学下单圈是如何分布的。这样，法瑞序列提供了一个深刻的联系数论，齐次动力学和几何，一个链接，应进一步推广和深化，以所有三个领域的利益。这个建议的目的是推广和深化这种联系，预期的结果将属于三个领域，倍增他们的利益。我们将研究类似的法瑞序列在非常一般的设置，如空间来自局部紧豪斯多夫（拓扑）群和他们的离散子群。这些可能是巨大而复杂的空间。拓扑群是这样的空间，我们有乘法的概念，即任何两个元素相乘在一起产生另一个元素在组中，并满足一些合理的规则。局部紧和豪斯多夫是两个拓扑概念，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。首先是研究它们的数论性质与经典的Farey序列和丰富的数论来自他们的类比。第二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horocycles。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，调和分析是一个数学领域，涉及将函数分解为“波函数”的无限和。通过推广以法瑞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。

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.