Fraenkel's Partition and Brown's Decomposition
| dc.creator | O'Bryant, Kevin | |
| dc.date | 2003-05-09 | |
| dc.date.accessioned | 2026-07-07T04:57:52Z | |
| dc.date.available | 2026-07-07T04:57:52Z | |
| dc.description | 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. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/math/0305133 | |
| dc.identifier | http://arxiv.org/abs/math/0305133 | |
| dc.identifier | Integers (2003), paper A11 (17 pages). http://www.integers-ejcnt.org/ | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67413 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11B83; 68R15 | |
| dc.title | Fraenkel's Partition and Brown's Decomposition | |
| dc.type | text |