Hypercubic billiard word: Difference between revisions
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=== Determining whether a scale word is a billiard scale === | === Determining whether a scale word is a billiard scale === | ||
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== Questions == | == Questions == | ||
1. A circular word ''s'' is ''d-balanced'' if for any ''k'' ≥ 1 and for any pair of length-''k'' subwords ''w'' and <i>w'</i> of ''s'', | 1. A circular word ''s'' is ''d-balanced'' if for any ''k'' ≥ 1 and for any pair of length-''k'' subwords ''w'' and <i>w'</i> of ''s'', | ||
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<math> \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d,</math> | <math> \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d,</math> | ||
where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. Is it the case that for all ''r'' ≥ 1, all ''r''-ary billiard circular words (''r'' − 1)-balanced? (The answer is known to be "yes" for ''r'' = 1, 2.) | where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. Is it the case that for all ''r'' ≥ 1, all ''r''-ary billiard circular words are (''r'' − 1)-balanced? (The answer is known to be "yes" for ''r'' = 1, 2.) | ||
2. Must ternary billiard scales have maximum variety at most 4, the MV4 ones being exactly the ones of balance 2? Does billiardness impose a deterministic relationship between arity, maximum variety and balance? | 2. Must ternary billiard scales have maximum variety at most 4, the MV4 ones being exactly the ones of balance 2? Does billiardness impose a deterministic relationship between arity, maximum variety and balance? | ||