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| For theorems relating to the GO property, see [[generator-offset property]]. | | For theorems relating to the GO property, see [[generator-offset property]]. |
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| ==== Definition: Billiard scale ====
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| Let
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| * ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub>, ..., ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many X<sub>''i''</sub> steps);
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| * ''n'' = ''a''<sub>1</sub> + ... + ''a''<sub>''r''</sub> be the length of ''w'';
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| * ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''<sub>1</sub> = 0 at a point (0, α<sub>1</sub>, α<sub>2</sub>, ... α<sub>''r''-1</sub>) where α<sub>''i''</sub> and α<sub>''j''</sub>/α<sub>''i''</sub> for ''i'' ≠ ''j'' are irrational.
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| We say that ''w'' is a ''billiard scale'' if any line in generic position of the form (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + ''v''<sub>0</sub> has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.
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| [[Category:Fokker block]] | | [[Category:Fokker block]] |