Inhomogenous approximation on manifolds and more general structures.

丢芬图近似是数论的一个领域，起源于有理数如何“快速”逼近实数的问题。近似的“速度”或“误差”是根据有理近似的分母的大小来衡量的。这一系列问题可以追溯到古希腊人和中国人，他们使用圆周率的良好有理近似值（3.14159…）来准确预测行星和恒星的位置。同样地，丢番图近似是对任意实数与有理数任意接近这一事实的定量分析；也就是说，有理数在实线上是密集的。丢芬图近似的度量理论是从测度论（概率论）的观点出发，用有理数研究实数的近似性质。中心主题是确定给定的近似性质是否在除测度0的特殊集合之外的任何地方都成立。在1924年的开创性工作中，Khintchine根据勒贝格测度建立了一个优雅的概率准则（“零满”定律），用于用有理数近似具有任意递减误差的实数。误差是有理近似的分母大小的函数，并且随着分母大小的增加而减小。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。丢番图近似的度量理论对于减小误差函数是完整的——Khintchine判据的类似物以及更精确和精细的Hausdorff度量理论陈述已经建立。现在假设n维空间中的点被限制在一个适当的子流形上；例如，二维空间中的曲线。这种限制意味着兴趣点在功能上是相关的（即变量是相关的），这就引入了主要的困难。直到最近，流形上丢番图近似的度量理论还局限于流形的特殊类别。在过去的十年中，主要受Kleinbock & Margulis开创性工作的影响，他们在1996年建立了Baker-Sprindzuk的基本“极值”猜想。实质上，已经建立了Khintchine在流形上的对偶逼近准则和在平面曲线上的同时逼近准则的Hausdorff测度类似物。虽然这构成了显著的进步，但流形理论还远未完成。当有理点或超平面被一个给定的量（非齐次因子）移动时，我们知道的很少。提出的研究的主要目的是解决这种不平衡，并在流形上发展一个非齐次近似的相干度量理论，使其达到与齐次近似相同的理解水平。开始和主要目标是研究流形的“非齐次极值”。一种新颖的思想是发展齐次极值与非齐次极值之间的转换技术。主要的成果将是一个定理，它对非齐次理论的作用就像Kleinbock和Margulis定理对齐次理论的作用一样。

Diophantine approximation is an area of number theory that originated with the question of how `rapidly' a real number can be approximated by rational numbers. The `speed' or `error' of approximation is measured in terms of the size of the denominator of the rational approximate. This line of questioning dates back to the ancient Greeks and Chinese who used good rational approximates to the number pi (3.14159...) in order accurately to predict the position of planets and stars. Equivalently, Diophantine approximation is a quantitative analysis of the fact that any real number is arbitrarily close to rational numbers; i.e. the rationals are dense in the real line.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-full' law) in terms of Lebesgue measure for a real number to be approximable by rationals with an arbitrary decreasing 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. 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. The metric theory of Diophantine approximation is complete for decreasing error functions -- the analogues of Khintchine's criterion have been established as well as the more precise and delicate Hausdorff measure theoretic statements. Now suppose that the points in n-dimensional space are restricted to lie on a proper submanifold; e.g. a curve in two-dimensional space. This restriction means that the points of interest are functionally related (i.e. the variables are dependent) and this introduces major difficulties. Until recently, the metric theory of Diophantine approximation on manifolds had been limited to special classes of manifolds. Over the last decade progress has been dramatic, mainly influenced by the pioneering work of Kleinbock & Margulis who in 1996 established the fundamental `extremality' conjecture of Baker-Sprindzuk. Essentially, the Hausdorff measure analogues of Khintchine's criterion for dual approximation on manifolds and simultaneous approximation on planar curves have now been established. Although this constitutes remarkable progress, the theory for manifolds is far from complete. When the rational points or hyperplanes are shifted by a given quantity (the inhomogeneous factor) very little is known. The main objective of the proposed research is to address this imbalance and develop a coherent metric theory of inhomogeneous approximation on manifolds to the same level of understanding as the one of homogeneous approximation. The starting and principle goal is to investigate `inhomogeneous extremality' for manifolds. A novel idea is to develop a transfer technique between homogeneous and inhomogeneous extremality. The major outcome will be a theorem that will be to the inhomogeneous theory what the Kleinbock & Margulis theorem has been to the homogenous theory.