Ternary scale theorems: Difference between revisions

Line 268:

==== Proof ====

For 7.1.1: ~~By previous results it~~ suffices to show that ternary balanced words are pairwise-MOS. 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.

For 7.1.1: It suffices to show that ternary balanced words are pairwise-MOS and that non-PWF pairwise-MOS scales are even-regular.

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.

For 7.1.2: Suppose ''s'' is balanced and has three sizes for ''k''-steps, ''a''''i'''''X''' + ''b''''i'''''Y''' + ''c''''i'''''Z''' = (''a''i, ''b''i, ''c''i) for ''i'' = 1, 2, 3. We may assume (''a''2, ''b''2, ''c''2) = (''a''1, ''b''1 + 1, ''c''1 - 1). Then either (''a''3, ''b''3, ''c''3) = (''a''1 + 1, ''b''1, ''c''1 - 1) or (''a''3, ''b''3, ''c''3) = (''a''1 - 1, ''b''1 + 1, ''c''1). In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes.

@@ Line 268: / Line 268: @@
 ==== Proof ====
-For 7.1.1: By previous results it suffices to show that ternary balanced words are pairwise-MOS. 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.
+For 7.1.1: It suffices to show that ternary balanced words are pairwise-MOS and that non-PWF pairwise-MOS scales are even-regular.
+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.
 For 7.1.2: Suppose ''s'' is balanced and has three sizes for ''k''-steps, ''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 ''i'' = 1, 2, 3. We may assume (''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 (''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 (''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.

Ternary scale theorems: Difference between revisions

Inthar (talk | contribs)