Ternary scale theorems: Difference between revisions
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==== Proof ==== | ==== Proof ==== | ||
For 7.1.1: We showed previously that the Fraenkel, odd-regular, and even-regular circular words are balanced. Thus it remains to show that (a) ternary balanced words are pairwise-MOS (b) if ''a'' > ''b'' > ''c'', then ''s'' is equivalent to the Fraenkel word (c) assuming ''a'' != ''b'' = ''c'' any ''s'' that is not odd-regular or even-regular | For 7.1.1: We showed previously that the Fraenkel, odd-regular, and even-regular circular words are balanced. Thus it remains to show that (a) ternary balanced words are pairwise-MOS (b) if ''a'' > ''b'' > ''c'', then ''s'' is equivalent to the Fraenkel word (c) assuming ''a'' != ''b'' = ''c'' any ''s'' that is not odd-regular or even-regular is not balanced. | ||
(a) Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y'''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property. | (a) Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y'''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property. | ||
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“_'''XYZXYXZXYX'''_”. Both _s are necessarily '''Z'''s. To end the proof, note that we have obtained the configuration around every '''Z''' and this determines the whole sequence. Thus ''W'' = ('''XYXZXYX''')<sup>ω</sup>. | “_'''XYZXYXZXYX'''_”. Both _s are necessarily '''Z'''s. To end the proof, note that we have obtained the configuration around every '''Z''' and this determines the whole sequence. Thus ''W'' = ('''XYXZXYX''')<sup>ω</sup>. | ||
(c) The scale made by taking ''s'' and conflating '''Y''' and '''Z''' into the letter '''W''' must be a MOS. To this scale we may imagine substituting a scale made of an equal amount of '''Y''' and '''Z''' letters into the "slot letters" '''W''' letter by letter. | (c) The scale made by taking ''s'' and conflating '''Y''' and '''Z''' into the letter '''W''' must be a MOS. To this scale we may imagine substituting a scale made of an equal amount of '''Y''' and '''Z''' letters into the "slot letters" '''W''' letter by letter. Let ''t''<sub>1</sub> be a length-''k'' subword of the form '''YX'''<sup>''k''-2</sup>'''Y''' under the projection. We may assume that the chunk sizes of the MOS are ''k'' - 2 and ''k'' - 1, or ''k'' - 2 and ''k'' - 3. Either way, there exists some subword with (''k'' - i)-many '''X'''s, i = 1 or 2, and two '''Z'''s. This violates balance because ''t''<sub>1</sub> contains zero '''Z'''s. | ||
For 7.1.2: Suppose ''s'' is balanced and has at least three sizes for ''k''-steps, {{nowrap|''a''<sub>''i''</sub>'''X''' + ''b''<sub>''i''</sub>'''Y''' + ''c''<sub>''i''</sub>'''Z''' {{=}} (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>, ''c''<sub>''i''</sub>)}} for {{nowrap|''i'' ∈ {{(}}1, 2, 3{{)}}}}. We may assume {{nowrap|(''a''<sub>2</sub>, ''b''<sub>2</sub>, ''c''<sub>2</sub>) {{=}} (''a''<sub>1</sub>, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub> − 1)}}. Then either {{nowrap|(''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) {{=}} (''a''<sub>1</sub> + 1, ''b''<sub>1</sub>, ''c''<sub>1</sub> − 1)}} or {{nowrap|(''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) {{=}} (''a''<sub>1</sub> − 1, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub>)}}. In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes. | For 7.1.2: Suppose ''s'' is balanced and has at least three sizes for ''k''-steps, {{nowrap|''a''<sub>''i''</sub>'''X''' + ''b''<sub>''i''</sub>'''Y''' + ''c''<sub>''i''</sub>'''Z''' {{=}} (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>, ''c''<sub>''i''</sub>)}} for {{nowrap|''i'' ∈ {{(}}1, 2, 3{{)}}}}. We may assume {{nowrap|(''a''<sub>2</sub>, ''b''<sub>2</sub>, ''c''<sub>2</sub>) {{=}} (''a''<sub>1</sub>, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub> − 1)}}. Then either {{nowrap|(''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) {{=}} (''a''<sub>1</sub> + 1, ''b''<sub>1</sub>, ''c''<sub>1</sub> − 1)}} or {{nowrap|(''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) {{=}} (''a''<sub>1</sub> − 1, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub>)}}. In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes. | ||