The Inhomogeneous Duffin-Schaeffer Conjecture

丢芬图近似——这个项目的主要关注点——是数论的一个领域，简单来说，它研究实数的有理数近似，即两个整数的分数的近似。它可以追溯到古希腊人和中国人，他们使用圆周率的合理近似值（=3.1415…）来预测行星和恒星的位置。在“日常生活”中，我们经常使用截断的十进制展开，例如3.14=314/100来近似圆周率。然而，这些通常都不太好。例如，22/7使用的位数比314/100少，但更接近圆周率，而355/113使用的位数相同，但准确地给出了圆周率的小数点后5位。狄利克雷基本定理保证了良好的有理数近似：对于每一个无理数x，都有无限多个有理数a/q逼近x，其范围在分母q的平方1分以内。当然，与所有实数相反，单个实数在如何逼近方面可能会有很大的不同。例如，刘维尔数通常可以用有理数a/q无限逼近到分母的任意次幂1以内，而对于难以逼近的数，幂只能是2，如狄利克雷定理。度量数论采用概率观点，因此在“研究所有”和“研究单个”数字之间提供了一个中间立场。该理论的中心主题是确定是否几乎所有实数都可以用有理数以某种方式近似。1941年，Duffin和Schaeffer提出了一个非常普遍的猜想，预测了几乎所有（在概率方面）实数都可以用有理数近似。解决这一猜想的尝试有着悠久的历史，并在此过程中有许多发现。这个猜想最终在Koukoulopoulos和Maynard的突破中得到了证明，其重要性获得了2022年菲尔兹奖（Fields Medal Award）的认可。这个项目将研究这个猜想的更一般的非齐次版本。在非齐次逼近中，有理数的分子被一个固定的实参数——非齐次部分平移。其原因最好用圆的旋转来描述。在齐次情况下，如果a/q近似于实数x，给定圆上的任何一点旋转q次的角度=2。X返回到其原始位置的邻域，其大小由近似误差决定。在非齐次情况下，这样的旋转被用来击中与非齐次部分相关的圆上任意不动点的邻域。本项目将开发一种新的方法来解决非齐次Duffin-Schaeffer猜想。特别地，我们的目标是发现满足猜想的非齐次部分的第一个非理性例子。由于它的概率性质，这个猜想毫不奇怪地被用第二个Borel-Cantelli引理的一个版本来处理。这使我们能够确定，如果我们假设事件具有一定的独立性，那么某一系列“发散”的“事件”就会以正概率无限频繁地发生。验证后者是解决问题的关键，因此构成了本项目的核心。特别是，我们将研究如何使用初始事件的变化来改进现有的和获得新的独立估计。这将汇集丢番图近似，数论和概率论的技术和思想。特别是，我们将在概率论中开发新的工具，这些工具将纳入0 - 1定律，使一个能够将正概率扩展到全概率。这个项目的主题可以在数论和动力系统等其他领域的许多其他问题中找到。因此，我们期望我们将开发的新技术和想法将对所涉及的领域和远远超出的领域产生持久的影响。

Diophantine approximation - the main concern of this project - is an area of number theory which, in simple terms, studies rational approximations to real numbers, that is approximations by fractions of two integers. It dates back to the ancient Greeks and Chinese who used good rational approximations to the number pi (=3.1415...) to predict the position of planets and stars. In 'everyday life' we often use truncated decimal expansions for the purpose, e.g. 3.14=314/100 to approximate pi. However, these are usually far from being good. For example, 22/7 uses fewer digits than 314/100 but is closer to pi, while 355/113 uses the same number of digits but accurately gives 5 decimal places of pi.Good rational approximations are guaranteed by Dirichlet's fundamental theorem: for every irrational number x there are infinitely many rationals a/q approximating x to within 1 over the square of the denominator, q. Of course, individual real numbers, as opposed to all real numbers, may vary vastly in terms of how they can be approximated. For instance, Liouville numbers can be approximated infinitely often by rationals a/q to within 1 over any power of the denominator, while for badly approximable numbers that power can only be 2, as in Dirichlet's theorem. Metric number theory takes a probabilistic viewpoint and thus offers a middle ground between 'studying all' and 'studying individual' numbers. The central theme of this theory is to determine whether almost all or almost no real numbers can be approximated by rational numbers in a certain way.In 1941 Duffin and Schaeffer stated a very general conjecture predicting how almost all (in probabilistic terms) real numbers can be approximated by rational numbers. Attempts to solve the conjecture have a long history and many discoveries along the way. The conjecture was eventually proved in a breakthrough by Koukoulopoulos and Maynard, which magnitude was recognised by a 2022 Fields Medal Award to Maynard. This project will investigate the far more general inhomogeneous version of the conjecture.In inhomogeneous approximations the numerator of the rational number is shifted by a fixed real parameter - the inhomogeneous part. The reason for that is best described in terms of circle rotations. In the homogeneous case, if a/q approximates a real number x, any point on a given circle rotated q times by the angle alpha=2.pi.x returns to a neighborhood of its original position, which size is determined by the error of approximations. In the inhomogeneous case such rotations are used to hit the neighborhood of an arbitrary fixed point on the circle associated with the inhomogeneous part.This project will develop a novel approach to the inhomogeneous Duffin-Schaeffer conjecture. In particular, we aim to discover the first irrational examples of the inhomogeneous part satisfying the conjecture. For its probabilistic nature the conjecture is unsurprisingly treated using a version of the second Borel-Cantelli lemma. This enables one to establish that a certain 'divergent' series of 'events' happens infinitely often with positive probability if we assume a ceratin independence of the events. Verifying the latter is the key to solving the problem and thus constitutes the core of this project. In particular, we will investigate how variations of the initial events can be used to improve existing and obtain new independence estimates. This will bring together techniques and ideas from Diophantine approximation, number theory and probability. In particular, we will develop novel tools in probability theory that will incorporate zero-one laws enabling one to extend positive to full probabilities. The theme of this project can be found in many other problems in number theory and other areas such as dynamical systems. Thus, we expect that the novel techniques and ideas that we will develop will have a lasting impact on the areas involved and far beyond.