Circle rotations and their generalisations in Diophantine approximation

丢番图逼近是研究如何以及真实的数字可以近似有理数。在整个数学史上，这一直是应用于真实的世界问题的最重要领域之一。今天，丢番图近似用于模拟科学实验和其他自然行为的数值算法和计算机程序。丢番图近似在许多其他数学和科学环境中也起着重要的作用，作为结果的支撑结构。丢番图近似中有几个长期存在的开放问题，最近引起了更广泛的数学界的关注。其中之一是利特尔伍德猜想，它预测了真实的数对可以同时被具有相同分母的有理数近似。本课题的目的是利用圆旋转的分布及其推广来研究Littlewood猜想及其相关问题。假设你取一个周长为1的圆，并在边界的沿着某处挑出一点。如果你将整个圆旋转一个固定的角度，你的点将移动到圆上的一个新位置。如果你考虑无限次重复这个旋转，那么点的所有可能位置的集合就叫做它的轨道。了解轨道的点在一个给定的旋转是一个基本的问题，这是直接关系到了解如何以及一个真实的数可以近似的fractions.I最近表明了如何一种技术称为奥斯特洛夫斯基展开可以用来证明大量的新结果关于Littlewood猜想。Ostrowski展开基本上允许我们将点的轨道重新组织成一个无限的块阵列，每个块都可以通过使用数论技术来理解。通过这种方式，奥斯特洛夫斯基展开可以用来分离出利特尔伍德猜想中的一个变量，从而将这个问题转换到一维的环境中。对圆旋转的理解很可能会导致整个利特尔伍德猜想的证明。然而，也有其他几个有趣的问题，这是开放的攻击，通过这种方法。这样的问题，我将调查被称为“收缩目标”的问题。在这里，你考虑一个圆旋转和每个元素的轨道上的一个点，你附加一个小球的一定半径。球的半径应该随着旋转的进行而缩小，问题是确定圆上的哪些点被无限多个球捕获。在这里提出的形式，这个问题的答案是已知的。然而，它仍然是一个开放的问题，证明一个定量的结果，这将告诉我们的比例球捕获一个给定的点上的圆圈。这类问题在动力学系统和粒子物理学中有着重要的意义。另一个问题是用圆的另一种变换来代替圆的旋转。所谓的“区间交换变换”是圆旋转的推广，它与丢番图逼近和动力系统中的问题有关。可以将这些变换中的每一个与编码关于点的轨道的信息的奥斯特洛夫斯基展开相关联。这样的框架，我们正在开发的研究利特尔伍德猜想也应该让我们证明新的和有趣的结果在许多设置。

Diophantine approximation is the study of how well real numbers can be approximated by rational numbers. Throughout the history of mathematics this has been one of the most important fields in applications to real world problems. Today Diophantine approximation is used in numerical algorithms and computer programs which model scientific experiments and other natural behaviour. It also plays a significant role as a supporting structure for results in many other mathematical and scientific settings.There are several long standing open problems in Diophantine approximation which have attracted recent attention in the wider mathematical community. One of these is the Littlewood Conjecture, which predicts how well pairs of real numbers can be simultaneously approximated by rationals with the same denominator. The goal of this project is to investigate the Littlewood Conjecture and related problems by using information about the distribution of circle rotations and their generalisations.Suppose you take a circle of circumference one and single out a point somewhere along the boundary. If you rotate the whole circle through a fixed angle your point will move to a new position on the circle. If you think about repeating this rotation infinitely many times then the collection of all possible positions of the point is called its orbit. Understanding the orbits of points under a given rotation is a basic problem which is directly related to understanding how well a real number can be approximated by fractions.I have recently shown how a technique called Ostrowski expansion can be used to prove substantial new results about the Littlewood Conjecture. Ostrowski expansion basically allows us to reorganize the orbits of points into an infinite array of blocks, each of which can then be understood by using number theoretic techniques. In this way the Ostrowski expansion can be used to isolate one of the variables in the Littlewood Conjecture and thereby recast the problem in a one-dimensional setting.This understanding of circle rotations may well lead to the proof of the entire Littlewood Conjecture. However there are also several other interesting problems which are open to attack via this method.One such problem which I will investigate is known as the "shrinking targets" problem. Here you consider a circle rotation and to each element in the orbit of a point you attach a small ball of a certain radius. The radii of the balls should shrink as the rotation progresses, and the problem is to determine which points on the circle are captured in infinitely many of the balls. In the form presented here the answer to this problem is known. However it is still a wide open problem to prove a quantitative result, which would tell us something about the proportion of balls which capture a given point on the circle. These types of problems have consequences in dynamical systems and particle physics.Another problem is to replace the circle rotation by a different transformation of the circle. The so-called "interval exchange transformations" are generalisations of circle rotations which are relevant to problems in Diophantine approximation and dynamical systems. It is possible to associate to each of these transformations an Ostrowski expansion that encodes information about the orbits of points. In this way the framework which we are developing to study the Littlewood Conjecture should also allow us to prove new and interesting results in many settings.