论文标题
对单位正方形的定量等分分配的研究
A study in quantitative equidistribution on the unit square
论文作者
论文摘要
在数学的不同领域，包括代数几何和差异理论，已经考虑了单位平方平移流动流的分布性能。量化等分分配的一种方法是比较特定集中花费的转换流量$ e \ subset [0,1]^2 $与预期时间之间的错误。在本文中，我们证明，当由凸面生成的代数中$ e $时，错误最多是$ \ log（t）^{1+ \ varepsilon} $，除了很多方向外，所有方向都有很多。每当方向差不多时，界限都可以锐化为$ \ log（t）^{1/2+\ varepsilon} $。我们产生的错误估计值比贝克证明的一般可测量集要小，而我们的示例类别大于Grepstad-Larcher的工作，后者获得了其集合的限制剩余属性。我们的证明依赖于边界的局部凸与流量的规则性之间的二元性。
The distributional properties of the translation flow on the unit square have been considered in different fields of mathematics, including algebraic geometry and discrepancy theory. One method to quantify equidistribution is to compare the error between the actual time the translation flow spent in specific sets $E \subset [0,1]^2$ to the expected time. In this article, we prove that when $E$ is in the algebra generated by convex sets the error is of order at most $\log(T)^{1+\varepsilon}$ for all but countably many directions. Whenever the direction is badly approximable the bound can be sharpened to $\log(T)^{1/2+\varepsilon}$. The error estimates we produce are smaller than for general measurable sets as proved by Beck, while our class of examples is larger than in the work of Grepstad-Larcher who obtained the bounded remainder property for their sets. Our proof relies on the duality between local convexity of the boundary and regularity of sections of the flow.
