Fraenkel分解与Brown分解
Fraenkel's Partition and Brown's Decomposition
摘要：
用B（x，x '）表示序列（[（n-x'）/ x ]）_{n=1}^ \infty，即所谓的Beatty序列。Fraenkel分割定理给出了B（x，x '）和B（y，y'）为正整数的充要条件，即对于B（x，x ′）\cap B（y，y ′）= \emptyset和B（x，x ′）\cup B（y，y ′）= {1，2，3，.}。固定0 < x < 1，如果k \in B（x，0），则c_k = 1，否则c_k = 0，即，c_k=[（k+1）/ x ] - [ k / x]。对于正整数m，令C_m为二进制字c_1c_2c_3. c_m。布朗分解给出整数q_1，q_2，.，与m无关并且至少指数增长，以及整数t，z_0，z_1，z_2，.，z_t（取决于m），使得C_m = C_{q_t} ^{z_t}C_{ q_{t-1} }^{ z_{t-1} }. C_{q_1} ^{z_1} C_{q_0} ^{z_0}。换句话说，布朗分解给出了C_ \infty的初始段的稀疏集合和C_m（对于每一个m）到这些初始段的乘积的显式分解。
英文摘要：
Denote the sequence ([ (n-x') / x ])_{n=1}^\infty by B(x, x'), a so-called Beatty sequence. Fraenkel's Partition Theorem gives necessary and sufficient conditions for B(x, x') and B(y, y') to tile the positive integers, i.e., for B(x, x') \cap B(y, y') = \emptyset and B(x, x') \cup B(y, y') = {1,2, 3, ...}. Fix 0 < x < 1, and let c_k = 1 if k \in B(x, 0), and c_k = 0 otherwise, i.e., c_k=[ (k+1) / x ] - [ k / x]. For a positive integer m let C_m be the binary word c_1c_2c_3... c_m. Brown's Decomposition gives integers q_1, q_2, ..., independent of m and growing at least exponentially, and integers t, z_0, z_1, z_2, ..., z_t (depending on m) such that C_m = C_{q_t}^{z_t}C_{q_{t-1}}^{z_{t-1}} ... C_{q_1}^{z_1}C_{q_0}^{z_0}. In other words, Brown's Decomposition gives a sparse set of initial segments of C_\infty and an explicit decomposition of C_m (for every m) into a product of these initial segments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
