自动机序列或更一般的代换序列最初源于计算机科学。它们是最简单的非周期低复杂性的序列，可以有限步构造，因此在数学的许多不同分支、理论物理、理论计算机、信息论等领域均有重要应用。在本课题中，我们将从数论和遍历论的视角来研究这些序列的数学理论。我们将利用它们来构造有限域上的形式幂级数、p-进实数以及p-adic动力系统，讨论所构造的这些元素的连分式展式、超越性、代数无关性、无理指数、组合结构、遍历分布，进而考察它们在数论、调和分析、分形几何等方面的应用。在此基础上，我们将探索生成这些序列的有限自动机与它们的算术、遍历性质之间更为深刻的联系，以期在上述课题研究中取得突破性的进展。

Automatic sequences or more generally substitutive sequences originally arise from computer science. They are the simplest aperiodic sequences with low complexity and can be constructed in finite steps, thus have important applications in many different branches of mathematics, theoretical physics, theoretical computer science, information theory, and so on. In this proposal, we shall study the mathematical theory of these sequences from the viewpoint of number theory and ergodic theory. We shall use them to construct formal power series over finite fields, p-ary real number, and p-adic dynamical systems, discuss continued fraction expansions, transcendence, algebraic independence, irrationality exponents, combinatorial structures, ergodic distributions of these constructed elements, and investigate their applications in number theory, harmonic analysis, fractal geometry, etc. Based on the above study, we shall explore the deep relationship between finite automata which generate these sequences, and the arithmetic and ergodic properties of these sequences, in the hope to obtain a breakthrough in the study of the above subjects in discussion.