Ternary scale theorems: Difference between revisions
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* B.3. For a MOS binary word, the count of a given letter in a generator is coprime to the total count of that letter in one period of the MOS. By this fact applied to ''T'', {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}. | * B.3. For a MOS binary word, the count of a given letter in a generator is coprime to the total count of that letter in one period of the MOS. By this fact applied to ''T'', {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}. | ||
* B.4. Hence, since every instance of the generator in ''T'' has ''j''-many '''W''' letters, every instance of ''g''<sub>1</sub> and every instance of ''g''<sub>2</sub> has ''j''-many non-'''X''' letters. | * B.4. Hence, since every instance of the generator in ''T'' has ''j''-many '''W''' letters, every instance of ''g''<sub>1</sub> and every instance of ''g''<sub>2</sub> has ''j''-many non-'''X''' letters. | ||
* C.1. Importantly, deleting '''X''''s gives windows of length ''j'', such that when you project adjacent lifted generators (by deleting '''X''''s) to the binary necklace {{nowrap|''U'' = ''E''<sub>'''X'''</sub>(''w'')('''Y''', '''Z''')}}, the resulting ''j''-step windows in ''U'' are adjacent and do not overlap. | * C.1. Importantly, deleting '''X''''s gives windows of length ''j'', such that when you project adjacent lifted generators (by deleting '''X''''s) to the binary necklace {{nowrap|''U'' :{{=}} ''E''<sub>'''X'''</sub>(''w'')('''Y''', '''Z''')}}, the resulting ''j''-step windows in ''U'' are adjacent and do not overlap. | ||
* C.2. Moreover, for every ''j''-step window {{nowrap|''U''[''q'' : ''q'' + ''j'']}}, there exists an {{nowrap|(''i'' + ''j'')-step}} window {{nowrap|''w''[''r'' : ''r'' + ''i'' + ''j'']}}, so that {{nowrap|''w''[''r'']}} is a non-'''X'''. Since by subclaim A, the unique imperfect {{nowrap|(''i'' + ''j'')-step}} window in ''w'' begins in an '''X''', we know that {{nowrap|''w''[''r'' : ''r'' + ''i'' + ''j'']}} is perfect. | * C.2. Moreover, for every ''j''-step window {{nowrap|''U''[''q'' : ''q'' + ''j'']}}, there exists an {{nowrap|(''i'' + ''j'')-step}} window {{nowrap|''w''[''r'' : ''r'' + ''i'' + ''j'']}}, so that {{nowrap|''w''[''r'']}} is a non-'''X'''. Since by subclaim A, the unique imperfect {{nowrap|(''i'' + ''j'')-step}} window in ''w'' begins in an '''X''', we know that {{nowrap|''w''[''r'' : ''r'' + ''i'' + ''j'']}} is perfect. | ||
* C.3. Also note that we only need to stack {{nowrap|2''b'' ≤ ''n'' − 1}} generators to witness this alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the 2''b''-th ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''. | * C.3. Also note that we only need to stack {{nowrap|2''b'' ≤ ''n'' − 1}} generators to witness this alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the 2''b''-th ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''. | ||