The Inhomogeneous Duffin-Schaeffer Conjecture

非齐次达芬-谢弗猜想

• 批准号: EP/X030784/1
• 负责人: Victor Beresnevich
• 金额: $ 10.32万
• 依托单位: University of York
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX030784%2F1
• 关键词: Inhomogeneous; Duffin; Schaeffer; Conjecture

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.

丢番图逼近-这个项目的主要关注点-是数论的一个领域，简单地说，它研究真实的数的有理逼近，即两个整数的分数的逼近。它可以追溯到古希腊人和中国人谁使用良好的合理近似数pi（=3.1415.）来预测行星和恒星的位置在“日常生活”中，我们经常使用截断小数展开，例如3.14=314/100来近似π。然而，这些通常都不是很好。例如，22/7比314/100使用更少的位数，但更接近pi，而355/113使用相同的位数，但精确地给出了pi的5个小数位。狄利克雷基本定理保证了良好的有理逼近：对于每个无理数x，有无穷多个有理数a/q在分母q的平方上逼近x到1以内。当然，与所有真实的数字相反，单个真实的数字在如何近似方面可能会有很大的不同。例如，刘维尔数可以被有理数a/q无限逼近到分母的任何幂的1以内，而对于不好逼近的数，幂只能是2，如狄利克雷定理。度量数论采用概率观点，因此提供了“研究所有”和“研究个体”数字之间的中间地带。该理论的中心主题是确定几乎所有或几乎没有真实的数是否可以以某种方式用有理数逼近。1941年，Duffin和Schaeffer提出了一个非常普遍的猜想，预测几乎所有（用概率术语）真实的数如何可以用有理数逼近。解决这个猜想的尝试有着悠久的历史和沿着的许多发现。这一猜想最终被Koukoulopoulos和Maynard证明，并获得了2022年菲尔兹奖。这个项目将研究更一般的非齐次版本的猜想。在非齐次近似的有理数的分子被移动了一个固定的真实的参数-非齐次部分。其原因最好用圆旋转来描述。在齐次情况下，如果a/q近似于真实的数x，则给定圆上的任何点旋转q次角度α = 2 π x返回到其原始位置的邻域，其大小由近似误差确定。在非齐次情形下，这种旋转被用来击中与非齐次部分相关联的圆上的任意不动点的邻域。本项目将为非齐次Duffin-Schaeffer猜想提供一种新的方法。特别是，我们的目标是发现第一个不合理的例子，满足猜想的非齐次部分。由于其概率性质，使用第二个博雷尔-坎特利引理的版本来处理该猜想并不令人惊讶。这使人们能够建立一个特定的“发散”系列的“事件”发生无限经常与积极的概率，如果我们假设一个certatin独立的事件。后者是解决问题的关键，因此构成了本项目的核心。特别是，我们将研究如何变化的初始事件可以用来改善现有的，并获得新的独立性估计。这将汇集从丢番图近似，数论和概率的技术和思想。特别是，我们将开发新的工具，在概率论，将纳入0 - 1法律，使一个扩展积极的全概率。这个项目的主题可以在数论和其他领域（如动力系统）的许多其他问题中找到。因此，我们希望我们将开发的新技术和想法将对所涉及的领域产生持久的影响。