Fraenkel's Partition and Brown's Decomposition

Abstract

Denote the sequence ([ (n-x&#39;) / x ])_{n=1}^\infty by B(x, x&#39;), a so-called Beatty sequence. Fraenkel&#39;s Partition Theorem gives necessary and sufficient conditions for B(x, x&#39;) and B(y, y&#39;) to tile the positive integers, i.e., for B(x, x&#39;) \cap B(y, y&#39;) = \emptyset and B(x, x&#39;) \cup B(y, y&#39;) = {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&#39;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&#39;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.