Mathematical Sciences: Ergodic Theory and Applications in Combinatorial Number Theory

本文的中心主题是区间交换变换的丢番图性质。与其他人（H.Masur， W.Veech等）的“全局”方法相反，Boshernitan在研究et时的方法特别强调所涉及参数的丢芬图和代数性质可能影响et的动态性质的方式。解析数论中的许多结果可以推广到let’s。例如，Boshernitzan最近与C.Carroll联合得出的二次域上的最小let必须是伪anosof（自相似）的结果，可以看作是经典拉格朗日关于二次无理数的连分数表示的周期性的结果的扩展。也许在这个方向上最杰出的猜想来自于一种尝试，即把著名的关于代数无理数的速率的罗斯定理推广到有理数可以近似。（相应的刘维尔定理的类比是正确的。）Boshernitzan用来解决上述问题的工具之一是他最近发现了动力系统中典型轨道的递归率与空间的Hausdorff维之间的联系（庞加莱递归定理的定量版本）。对于我们来说，这种联系是双向的。在许多其他情况下也是如此（Y.Peres, d.o nstein, b.w weiss）。Boshernitzan的其他兴趣包括数论，多边形台球，沿整数子序列的遍历定理，Hardy域（与M.Wierdl， D.Berend， G.Kolesnik， I.Kornfeld和V.Bergelson合作的项目）。由于目前研究计划的基本性质，在这一点上很难准确地评估可能的应用领域。在可能从我们的研究中受益的主题中，我只提到在计算机技术中有用的新代码压缩方案的创建可能性（与我们发现的庞加莱递归定理的定量版本有关）和激光技术的可能应用（与I. Kornfeld关于区间平移映射的研究）。

Abstract Boshernitzan The central subject in this proposal is diophantine properties of iet's (interval exchange transformations). Boshernitan's approach in the study of iet's, contrary to the 'global' approaches of others (H.Masur, W.Veech etc.), makes special emphasis on the way diophantine and algebraic properties of the parameters involved may affect the dynamical properties of an iet. Many results in analytical number theory are extendable to iet's. For example, Boshernitzan's recent result, joint with C.Carroll, that minimal iet's over a quadratic field must be pseudo-Anosof (self-similar) can be viewed as an extension of the classical Lagrange's result on the periodicity of the continued fraction representations of quadratic irrationals. Perhaps the most outstanding conjecture in this direction is obtained from an attempt to extend to iet's the famous Roth's theorem on the rate algebraic irrationals can be approximated by the rationals. (The analogue of the corresponding Liouville's theorem is true.) One of the tools at Boshernitzan's disposal in tackling the above problems is his recent discovery of a connection between the rate of recurrence of typical orbits in dynamical systems and the Hausdorff dimension of the space (quantitative version of Poincare recurrence theorem). For iet's, this connection works both directions. The same is true in many other situations (Y.Peres, D.Ornstein, B.Weiss). The list of other Boshernitzan's interests includes number theory, billiards in polygons, ergodic theorems along subsequences of integers, Hardy fields (joint projects with M.Wierdl, D.Berend, G.Kolesnik, I.Kornfeld and V.Bergelson). Because of the basic nature of the present research proposal, it is hard at this point to assess accurately the fields of possible applications. Among the subject which may benefit from our research I would only mention the possibility of creation of new code compression schemes useful in computer technology ( related to our discovery of quantitative version of Poincare recurrence theorem) and possible applications to laser technology (research with I. Kornfeld on interval translation maps).