Classical metric Diophantine approximation revisited

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。这个理论可以追溯到古希腊人和中国人，他们对圆周率（3.14159...）使用了很好的合理近似。丢番图近似的度规理论是从测度论（概率）的角度研究真实的数的有理数近似性质的理论。中心主题是确定一个给定的近似属性是否在任何地方都成立，除了一个例外的零测量集。在他的开创性工作的1924年，欣钦建立了一个优雅的概率标准（一个'0 - 1'法）在勒贝格措施的一个真实的数是近似的理性与任意减少（单调）的错误。误差是有理逼近的扩张因子大小的函数，并随着扩张因子大小的增加而减小。单调性假设是至关重要的，因为否则该准则是错误的。在有理逼近被简化的自然假设下（即以其最低形式，使得在有理点处的逼近误差唯一确定），Duffin-Schaeffer猜想（1941）提供了适当的期望陈述，而没有单调性假设。它代表了数论中最著名的未解决的问题之一。一个主要目的是作出重大贡献，这一关键猜想利用最近的"鞅"的方法开发的海恩斯（命名的研究助理）和Vaaler。此外，我们还研究了用Hausdorff测度（一种分形量）代替Lebesgue测度的猜想的更一般形式。一个主要的成果将是Duffin-Schaeffer猜想的措施接近勒贝格措施。Duffin-Schaeffer猜想的重要性是毋庸置疑的。然而，它确实改变了欣钦考虑的问题的基本性质，因为理性近似被减少了。1971年，卡特林提出了一个关于无约束问题的猜想，其中的有理数不被假设为约简。卡特林声称他的猜想等价于Duffin-Schaeffer猜想。然而，他的证明包含了一个严重的缺陷，索赔本身仍然是一个有趣的问题。在更高的维度中，n维空间中任意点的有理点逼近（同时逼近）或有理超平面逼近（对偶逼近）是一维理论的自然推广。考虑一个线性形式的系统统一了两种形式，自然产生了线性形式理论。丢番图逼近的度量理论对于维数大于1的同时逼近是完备的。没有任何单调性假设的Khintchine准则（即同时Catlin猜想）和Duffin-Schaeffer猜想的类似物都已经建立，以及更精确和微妙的Hausdorff测度理论陈述。然而，对偶和更一般的线性形式理论还远未完成。在这个建议的线性形式类似物的Duffin-Schaeffer和Catlin结构精确制定。一个主要的目标是在大于一的维度上建立这些结构。一个新的想法是开发一个“切片”技术，减少了一个线性形式的问题，一个很好理解的同时问题。主要成果将是一个统一的线性形式理论在欧几里德空间。

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. The theory dates back to the ancient Greeks and Chinese who used good rational approximations to the number pi (3.14159...) in order to accurately predict the position of planets and stars.The metric 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 is to determine whether a given approximation property holds everywhere except on an exceptional set of measure zero. In his pioneering work of 1924, Khintchine established an elegant probabilistic criterion (a `zero-one' law) in terms of Lebesgue measure for a real number to be approximable by rationals with an arbitrary decreasing (monotonic) error. The error is a function of the size of the denominators of the rational approximates and decreases as the size of the denominators increases. The monotonicity assumption is crucial since the criterion is false otherwise. Under the natural assumption that the rational approximates are reduced (i.e. in their lowest form so that the error of approximation at a rational point is determined uniquely), the Duffin-Schaeffer conjecture (1941) provides the appropriate expected statement without the monotonicity assumption. It represents one of the most famous unsolved problems in number theory. A major aim is to make significant contributions to this key conjecture by exploiting the recent `martingale' approach developed by Haynes (the named Research Assistant) and Vaaler. Furthermore, a more general form of the conjecture in which Lebesgue measure is replaced by Hausdorff measure (a fractal quantity) will be investigated. A major outcome will be the Duffin-Schaeffer conjecture for measures close to Lebesgue measure. The importance of the Duffin-Schaeffer conjecture is unquestionable. However, it does change the underlying nature of the problem considered by Khintchine in that the rational approximates are reduced. In 1971, Catlin stated a conjecture for the unconstrained problem in which the rationals are not assumed to be reduced. Catlin claimed that his conjecture was equivalent to the Duffin-Schaeffer conjecture. However, his proof contained a serious flaw and the claim remains an interesting problem in its own right. In higher dimensions, the approximation of arbitrary points in n-dimensional space by rational points (simultaneous approximation) or rational hyperplanes (dual approximation) is the natural generalisation of the one-dimensional theory. Considering a system of linear forms unifies both forms and naturally gives rise to the linear forms theory. The metric theory of Diophantine approximation is complete for simultaneous approximation in dimension greater than one. The analogues of Khintchine's criterion without any monotonicity assumption (i.e. the simultaneous Catlin conjecture) and the Duffin-Schaeffer conjecture have both been established as well as the more precise and delicate Hausdorff measure theoretic statements. However, the dual and more generally the linear forms theory are far from complete. In this proposal the linear forms analogues of the Duffin-Schaeffer and Catlin conjectures are precisely formulated. A principle goal is to establish these conjectures in dimension greater than one. A novel idea is to develop a `slicing' technique that reduces a linear forms problem to a well understood simultaneous problem. The major outcome will be a unified linear forms theory in Euclidean space.