Circle rotations and their generalisations in Diophantine approximation

丢芬图近似是研究实数如何被有理数近似的方法。在整个数学历史中，这一直是应用于现实世界问题的最重要领域之一。今天，丢番图近似法被用于数值算法和计算机程序中，以模拟科学实验和其他自然行为。它还在许多其他数学和科学环境中作为支持结果的结构发挥着重要作用。丢番图近似中有几个长期存在的开放性问题，最近引起了数学界的广泛关注。其中之一是利特尔伍德猜想（Littlewood Conjecture），它预测了具有相同分母的有理数同时近似实数对的效果。这个项目的目标是通过使用关于圆旋转分布及其推广的信息来研究Littlewood猜想和相关问题。假设你取一个周长为1的圆，在边界上挑出一个点。如果你把整个圆旋转一个固定的角度，你的点就会移动到圆上的一个新位置。如果你想重复这个旋转无数次，那么这个点所有可能位置的集合就叫做它的轨道。理解给定旋转下点的轨道是一个基本问题，它直接关系到理解实数如何被分数近似。我最近展示了一种叫做Ostrowski展开的技术如何被用来证明Littlewood猜想的重要新结果。奥斯特洛夫斯基展开基本上允许我们将点的轨道重新组织成无限的块数组，每个块都可以通过使用数论技术来理解。通过这种方式，Ostrowski展开可以用来分离Littlewood猜想中的一个变量，从而在一维环境中重新定义问题。这种对圆旋转的理解很可能导致整个利特尔伍德猜想的证明。然而，也有一些其他有趣的问题，可以通过这种方法进行攻击。我将研究的一个这样的问题被称为“收缩目标”问题。这里你考虑一个圆的旋转，在一个点轨道上的每个元素上，你附加一个一定半径的小球。球的半径应该随着旋转的进行而缩小，问题是确定圆圈上的哪些点被无限多个球捕获。在这里给出的形式中，这个问题的答案是已知的。然而，证明一个定量的结果仍然是一个悬而未决的问题，这个结果将告诉我们一些关于球在圆上给定点上的比例。这类问题在动力系统和粒子物理学中都有影响。另一个问题是用圆的另一个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，它与丢番图近似和动力系统中的问题有关。有可能将这些变换中的每一个与奥斯特洛夫斯基展开联系起来，该展开对点的轨道信息进行编码。通过这种方式，我们正在开发的研究Littlewood猜想的框架也应该允许我们在许多情况下证明新的和有趣的结果。

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.