Interleaving: Difference between revisions
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If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals. | If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals. | ||
{{proof| Denote by |''w''| the length of subword ''w'' in letters and by ‖''w''‖ the interval subtended by subword ''w'' in its circular word. | {{proof|contents=Denote by |''w''| the length of subword ''w'' in letters and by ‖''w''‖ the interval subtended by subword ''w'' in its circular word. | ||
Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>. | Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>. | ||
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=== Attempted proof of Conjecture === | === Attempted proof of Conjecture === | ||
{{proof|Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''. | {{proof|contents=Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''. | ||
Case 1: gcd(2(''a'' + ''b''), ''k'') = 1. If ''k'' = 1, then any '''Z'''<sup>''q''</sup> a maximal subword of consecutive '''Z'''s has ''q'' odd, and they all must be the same length and separated by one non-'''Z''' letter '''W''': | Case 1: gcd(2(''a'' + ''b''), ''k'') = 1. If ''k'' = 1, then any '''Z'''<sup>''q''</sup> a maximal subword of consecutive '''Z'''s has ''q'' odd, and they all must be the same length and separated by one non-'''Z''' letter '''W''': | ||