Mathematical Sciences: Ergodic Theory and Applications in Combinatorial Number Theory

抽象的Boshernitzan 这个建议的中心主题是区间交换变换的丢番图性质。 Boshernitan的方法在iet的研究，相反的“全球”的方法，其他人（H.Masur，W.Veech等）， 特别强调了 所涉及的参数的丢番图和代数性质可能影响IET的动力学性质。解析数论中的许多结果可推广到等熵数论。 例如，Boshernitzan最近与C.卡罗尔联合得出的结论，即二次域上的最小iet必须是伪Anosof（自相似），可以看作是经典拉格朗日关于二次无理数的连分式表示的周期性的结果的推广。 也许在这个方向上最突出的猜想是从试图扩大到iet的著名的罗斯定理率代数无理数可以近似的有理数。(The相应的刘维尔定理的模拟是正确的。） 波舍尼赞处理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系统的Hausdorff维数之间的联系。 Poincare递推定理（Poincare recurrence theorem） 对于iET来说，这种连接是双向的。在许多其他情况下也是如此（Y.Peres，D.Ornstein，B.韦斯）。列表的其他Boshernitzan的利益包括一些理论，台球在多边形，遍历定理沿着序的整数，哈代领域（联合 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，目前很难准确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，我只想提到创建新代码的可能性 用于计算机技术的压缩方案 （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的区间平移映射）。

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).