Circle rotations and their generalisations in Diophantine approximation

丢番图逼近是研究实数用有理数逼近的程度。纵观数学史，这一直是实际问题应用中最重要的领域之一。今天，丢番图近似被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中。它还作为许多其他数学和科学环境中结果的支撑结构发挥着重要作用。丢番图近似中有几个长期悬而未决的问题，最近引起了更广泛的数学界的关注。其中之一是Littlewood猜想，它预测实数对可以同时被具有相同分母的有理数近似得多好。这个项目的目标是利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。假设你取一个圆周的圆，并在边界的某个地方挑出一个点。如果将整个圆旋转固定角度，您的点将移动到圆上的新位置。如果你考虑无限次地重复这种旋转，那么该点所有可能位置的集合称为它的轨道。了解点在给定旋转下的轨道是一个基本问题，它直接关系到了解用分数来逼近实数的程度。我最近展示了一种名为Ostrowski展开的技术如何用来证明关于Littlewood猜想的实质性新结果。奥斯托夫斯基展开式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，然后每个块都可以用数论技术来理解。这样，Ostrowski展开式可以用来分离Littlewood猜想中的一个变量，从而在一维环境中重塑问题。这种对圆旋转的理解很可能导致整个Littlewood猜想的证明。然而，还有其他几个有趣的问题，可以用这种方法来攻击。其中一个我将调查的问题被称为“缩小目标”问题。在这里，你考虑圆的旋转，并且在一个点的轨道上的每个元素上附加一个特定半径的小球。随着旋转的进行，球的半径应该会缩小，问题是要确定在无限多的球中捕获了圆上的哪些点。在这里给出的表格中，这个问题的答案是已知的。然而，要证明一个定量的结果仍然是一个很大的悬而未决的问题，它将告诉我们一些关于捕获圆周上给定点的球的比例。这类问题在动力系统和粒子物理中都有影响。另一个问题是用圆的不同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。可以将编码关于点的轨道的信息的Ostrowski展开与这些变换中的每一个相关联。这样，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.