Classical metric Diophantine approximation revisited

丢番图逼近是数论的一个分支，它可以松散地描述为对每个实数都可以被一个有理数任意逼近的性质的定量分析。这一理论可以追溯到古希腊人和中国人，他们对数字圆周率(3.14159)使用了良好的有理近似。为了准确地预测行星和恒星的位置。丢番图近似的度规理论是从测度论(概率)的角度研究实数的有理逼近性质。其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，除了在一组例外的零度量上。在他1924年的开创性工作中，金钦建立了一个优雅的概率准则(“零-一”定律)，根据勒贝格度量，实数可以用具有任意递减(单调)误差的有理数来逼近。误差是有理近似的分母大小的函数，并且随着分母的大小而减小。单调性假设是至关重要的，因为否则标准就是假的。在有理逼近约化的自然假设下(即以有理逼近的最低形式使得在有理点的逼近误差是唯一确定的)，Duffin-Schaeffer猜想(1941)提供了适当的预期陈述，而没有单调性假设。它代表了数论中最著名的悬而未决的问题之一。一个主要目的是通过利用Haynes(指定的研究助理)和Vaaler最近开发的“鞅”方法，对这一关键猜想作出重大贡献。此外，我们还将研究一种更一般形式的猜想，即用Hausdorff测度(一个分形量)代替勒贝格测度。一个主要结果将是Duffin-Schaeffer猜想，即关于接近勒贝格度量的度量。杜芬-谢弗猜想的重要性是毋庸置疑的。然而，它确实改变了金钦所考虑的问题的根本性质，因为有理近似被减少了。1971年，Catlin提出了一个关于无约束问题的猜想，其中的有理数不被假定为减少。卡特林声称他的猜想等同于Duffin-Schaeffer猜想。然而，他的证明包含一个严重的缺陷，而这一说法本身仍然是一个有趣的问题。在高维空间中，用有理点(同时逼近)或有理超平面(对偶逼近)逼近n维空间中的任意点是一维理论的自然推广。考虑线性形式的系统，统一了这两种形式，自然地产生了线性形式理论。丢番图逼近的度规理论对于一维以上的同时逼近是完备的。在没有任何单调性假设(即同时Catlin猜想)和Duffin-Schaeffer猜想的情况下，已经建立了与Khintchine判据类似的结果，并得到了更精确、更精细的Hausdorff测度理论表述。然而，对偶理论，更广泛地说，线性形式理论还远远不完整。在这个建议中，精确地表述了Duffin-Schaeffer和Catlin猜想的线性形式类似物。一个主要目标是在大于一维的维度上建立这些猜想。一个新的想法是开发一种“切片”技术，将一个线性形式的问题简化为一个众所周知的同时问题。主要成果将是欧几里得空间中的统一线性形式理论。

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.