无穷字符集上自相似序列的相关分形及数论问题的研究

通过迭代、可取值于无穷字符集上的自相似序列是比自动机序列、k-正则序列更为广泛的序列，具有生成方式简单、可有限步构造、自相似等特点，是分形几何、数论、理论计算机和信息论等学科的交叉。本项目将涉及分形几何与动力系统、数论、丢番图逼近理论等方面，主要研究无穷字符集上自相似序列的分形性质及其相关动力系统问题；其次探索自相似序列所生成的sum-free集的算术性质，利用其构成集合的相似性、嵌套性和迭代关系计算相关分形维数；同时利用Padé逼近和Hankel行列式的计算，研究无穷字符集上自相似序列所对应的b-进实数的无理指数。本项目研究的问题相对独立又密切相关，旨在揭示无穷字符集上自相似序列的分形、动力系统和相关的数论性质，丰富无理指数的丢番图性质理论，进而将分形与数论的相关领域有效地结合起来，寻求解决此类问题的新方法，以期在本项目研究中取得突破性的进展。

The self-similar sequences generated by iterating, taking its values of.infinite alphabets are a more generalization of the concept of automatic sequences, k-regular sequences and so on. It has a simple generating method, can be constructed in finite steps and is of self-similarity. Moreover, it is the central concept at the intersection of fractal theory, number theory, theoretical computer science, information theory, and so on. This research involves fractal, number theory, Diophantine approximation theory and we shall mainly study the fractal properties of the self-similar sequences on infinite alphabets and their related dynamical system. In this project, we shall discuss the arithmetic properties of the sum-free sets generated by the self-similar sequences. We shall use the related sets of self-similarity, nesting and iteration relation to calculate its fractal dimensions. Meanwhile, we shall use Padé approximation and calculate the Hankel determinants to study the irrationality exponent of b-nary real numbers corresponding to the self-similar sequences on infinite alphabets. These problems are relatively independent but closely related. We shall explore the properties of the fractal and dynamical system of the self-similar sequences on infinite alphabets and related properties of number theory, enriching Diophantine theory of the irrationality exponent. In addition, our research tries to combine number theory with related fractal effectively and look for new methods to solve these problems, in the hope to obtain a breakthrough in the study of the above subjects in discussion.