New frameworks in metric Number Theory: foundations and applications

丢番图逼近是数论的一个分支，可以粗略地描述为对每个真实的数可以被任意接近的有理数逼近的性质的定量分析;即有理数在真实的线上是稠密的。这个理论可以追溯到古希腊人和中国人，他们对数字pi（3.14...）使用了很好的合理近似。以准确预测行星和恒星的位置。今天，该理论与其他数学领域如遍历理论、动力系统和分形几何深深交织在一起。它继续在应用于真实的世界问题中发挥重要作用，包括那些由电子通信、天线设计和信号处理等快速发展领域产生的问题。几个世纪以来，丢番图近似理论的发展取得了惊人的成就，使今天数学研究中的一些重大挑战具体化，特别是Littlewood和Duffin-Schaeffer的理论和广义的Baker-Schmidt问题。这些挑战构成了研究计划的支柱。简而言之，我们计划在度量丢番图近似中开发大胆的新框架，目标是解决具有挑战性和主题性的问题。丢番图逼近的度量理论是从测度论（概率）的角度研究真实的数的有理数逼近性质。这个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，除了一个例外的零度量集。利特尔伍德猜想（英语：Littlewood's Conjecture）是一个普适命题，它预测了真实的数对可以被具有相同分母的有理数乘法逼近的程度。将通用丢番图近似问题转化为“扭曲”概率问题是待开发的新框架之一的一个例子。从本质上讲，这将使我们能够使用的语言和机械发展的格律丢番图近似的问题，先验不与格律数论。这种新颖的重新表述，然后，将允许一个新的攻击一些长期存在的假设。例如，在Littlewood猜想的情况下，概率重新表述自然地产生了“扭曲”的标准（非齐次）理论的度量丢番图近似。即使这也是一片未开发的领域。在数学中，甚至在科学中，一个新的观点可以是解决一个老问题的关键，而且可以导致新理论的蓬勃发展。在过去，上述三个研究挑战被认为涉及不同的想法。然而，最近的进展表明，它们之间存在着实质性的联系。例如，固定Littlewood猜想中的一个真实的数使我们能够根据Duffin-Schaeffer猜想的一维设置（非单调近似的误差）来重铸问题。反过来，通过从根本上利用奥斯特洛夫斯基计数的数字，这导致了新的格律见解利特尔伍德的猜想。此外，将二维近似问题限制在一条直线（或更一般的曲线）上，自然会用到流形的丢番图近似理论。广义贝克-施密特问题是中央发展这一理论，是密切相关的关键问题，关于分配合理点附近的流形。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.