New frameworks in metric Number Theory: foundations and applications

丢番图逼近是数论的一个分支，它可以松散地描述为对每个实数都可以被一个有理数任意接近地逼近的性质的定量分析，即有理数在实直线上是稠密的。这一理论可以追溯到古希腊人和中国人，他们对数字圆周率(3.14…)使用了良好的有理近似。为了准确预测行星和恒星的位置。今天，这一理论与其他数学领域，如遍历理论、动力系统和分形几何深度交织在一起。它继续在现实世界问题的应用中发挥重要作用，包括电子通信、天线设计和信号处理等快速发展的领域所产生的问题。丢番图近似理论的发展在几个世纪里取得了令人瞩目的成就的同时，也明确了当今数学中的一些主要研究挑战；特别是利特尔伍德和达芬-谢弗的猜想以及推广的贝克-施密特问题。这些挑战构成了研究方案的支柱。简而言之，我们计划在度量丢番图近似下开发大胆的新框架，目标是解决具有挑战性的和热门的问题。丢番图逼近的度量理论是从测度论(概率论)的角度研究实数的有理逼近性质。这一理论的中心主题是确定一个给定的近似性质是否在任何地方都成立，除了在一组例外的零度量上。Littlewood的猜想预测了实数对可以多好地被具有相同分母的有理数乘法逼近，这是一个普遍的声明，因为相关的逼近性质要求对所有的点都成立。将普遍的丢番图近似问题转化为“扭曲的”概率问题就是有待开发的新框架之一。本质上，这将允许我们使用度量丢番图近似中开发的语言和机制来解决先验与度量数论没有联系的问题。因此，这种新颖的重新提法将允许对一些长期存在的猜测进行新的攻击。例如，在Littlewood猜想的情况下，通过‘扭曲’度量丢番图近似的标准(非齐次)理论，自然地产生了概率改写。即便如此，这也代表着未开发的领域。在数学中--甚至在科学中--一个新的观点可能是解决一个旧问题的关键，而且可能会导致新的理论蓬勃发展。在过去，上述三个研究挑战被认为涉及截然不同的想法。然而，最近的进展表明，它们之间存在着实质性的联系。例如，确定Littlewood猜想中的一个实数使我们能够根据Duffin-Schaeffer猜想的一维设置(非单调的逼近误差)来重塑问题。反过来，通过对数字的奥斯托夫斯基计数的基本使用，这导致了对利特尔伍德猜想的新的格律见解。此外，将二维近似问题限制在一条直线(或更一般的曲线)上，自然会发挥流形的丢番图逼近理论。推广的贝克-施密特问题是这一理论发展的核心，并与有关流形附近有理性点的分布的关键问题密切相关。利用研究挑战之间的联系是该方案的一个主要特点。

Diophantine approximation is a branch of number theory that can loosely be described as a quantitative analysis of the property that every real number can be approximated by a rational number arbitrarily closely; i.e. the rationals are dense in the real line. The theory dates back to the ancient Greeks and Chinese who used good rational approximations to the number pi (3.14...) in order to accurately predict the position of planets and stars. Today, the theory is deeply intertwined with other areas of mathematics such as ergodic theory, dynamical systems and fractal geometry. It continues to play a significant role in applications to real world problems including those arising from the rapidly developing areas of electronic communications, antenna design and signal processing. While yielding spectacular achievements over centuries, the development of the theory of Diophantine approximation has crystallised some of today's major research challenges in mathematics; in particular the conjectures of Littlewood and Duffin-Schaeffer and the generalised Baker-Schmidt problem. These challenges form the backbone of the research programme. In short, we plan to develop bold new frameworks in metric Diophantine approximation with the goal of solving challenging and topical problems. The metrical theory of Diophantine approximation is the study of the approximation properties of real numbers by rationals from a measure theoretic (probabilistic) point of view. The central theme of this theory is to determine whether a given approximation property holds everywhere except on an exceptional set of measure zero. Littlewood's Conjecture, which predicts how well pairs of real numbers can be multiplicatively approximated by rationals with the same denominator, is a universal statement in that the associated approximating property is required to hold for all points. Transforming universal Diophantine approximation problems into 'twisted' probabilistic problems is an example of one the novel frameworks to be developed. In essence this would allow us to use the language and machinery developed in metrical Diophantine approximation for problems that a priori are not connected with metrical number theory. This novel reformulation, then, would allow a fresh attack on some long-standing conjectures. For instance, in the case of Littlewood's Conjecture, the probabilistic reformulation arises naturally by 'twisting' the standard (inhomogeneous) theory of metrical Diophantine approximation. Even this represents unexplored territory. In mathematics - and indeed in science - a new viewpoint can be the key to solving an old problem and moreover can lead to flourishing new theories. In the past the three aforementioned research challenges were thought to involve disparate ideas. However recent advances have shown that there are substantial links between them. For example, fixing one of the real numbers in Littlewood's Conjecture enables us to recast the problem in terms of the one-dimensional setting of the Duffin-Schaeffer Conjecture (the error of approximation in non-monotonic). In turn, by making fundamental use of the Ostrowski numeration of numbers this has lead to new metrical insights into Littlewood's Conjecture. Also, restricting a two-dimensional approximation problem to a line (or more generally a curve) naturally brings into play the theory of Diophantine approximation of manifolds. The generalised Baker-Schmidt problem is central to the development of this theory and is intimately linked to key problems regarding the distribution of rational points near manifolds. The exploitation of the links between the research challenges is a key feature of the programme.