New frameworks in metric Number Theory: foundations and applications

丢番图近似是数论的一个分支，可以粗略地描述为对每一个实数都可以被一个有理数任意近似这一性质的定量分析；也就是说，有理数在实线上是密集的。这个理论可以追溯到古希腊人和中国人，他们使用圆周率（3.14…）的合理近似来准确预测行星和恒星的位置。今天，该理论与遍历理论、动力系统和分形几何等其他数学领域紧密交织在一起。它继续在应用于解决现实世界问题方面发挥重要作用，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。虽然几个世纪以来取得了惊人的成就，丢番图近似理论的发展已经明确了当今数学中一些主要的研究挑战；特别是Littlewood和Duffin-Schaeffer的猜想以及广义的Baker-Schmidt问题。这些挑战构成了研究项目的支柱。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，以解决具有挑战性和热门问题。丢芬图近似的格律理论是从测度论（概率论）的观点出发，用有理数来研究实数的近似性质。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，除了一个特殊的测度0集合。利特尔伍德猜想（Littlewood’s Conjecture）预测了实数对如何被具有相同分母的有理数相乘近似，这是一个普遍命题，因为相关的近似性质必须对所有点都成立。将普遍的丢番图近似问题转化为“扭曲的”概率问题是有待开发的新框架之一。从本质上讲，这将允许我们使用在韵律丢芬图近似中发展起来的语言和机制来解决先天与韵律数论无关的问题。因此，这种新颖的重新表述将对一些长期存在的猜想提出新的挑战。例如，在Littlewood猜想的例子中，概率的重新表述是通过“扭曲”标准的（非齐次的）丢番图近似理论而自然产生的。甚至这也代表着未开发的领域。在数学中——实际上在科学中——一个新的观点可能是解决老问题的关键，而且还可能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猜想中的一个实数，使我们能够根据Duffin-Schaeffer猜想（非单调近似误差）的一维设置来重新定义问题。反过来，通过对奥斯特洛夫斯基数法的基本运用，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为一条直线（或更一般地说是一条曲线），自然会使流形的丢番图近似理论发挥作用。广义Baker-Schmidt问题是这一理论发展的核心，并且与流形附近有理点分布的关键问题密切相关。利用研究挑战之间的联系是该计划的一个关键特征。

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.