Rank-3 scale theorems: Difference between revisions

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==== Definitions: Billiard scale ====

Let

* w be a scale word with signature a1X1, ..., arXr;

* ''w'' be a scale word with signature ''a''1X1, ..., ''a''''r''X''r'' (i.e. ''w'' is a scale word with ''a''''i''-many Xi steps);

* n = a1 + ... + ar be the length of w;

* ''n'' = ''a''1 + ... + ''a''''r'' be the length of ''w'';

* L be a line of the form L(t) = (a1, ..., ar)t + v0, where v0 is a constant vector in Rr.

* ''L'' be a line of the form ''L''(''t'') = (''a''1, ..., ''a''''r'')''t'' + '''v'''0, where '''v'''0 is a constant vector in '''R'''''r''.

We say that L is ''in generic position'' if L intersects the hyperplane x1 = 0 at a point (0, α1, α2, ... αr-1) where αi and αi/αj for i ≠ j are irrational.

We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''1 = 0 at a point (0, α1, α2, ... α''r''-1) where α''i'' and αj/αi for ''i'' ≠ ''j'' are irrational.

We say that ''w'' is a ''billiard scale'' if any appropriate line in generic position, (a1, ..., ar)t + v0, has intersections with coordinate level planes xi = k that spell out the scale as you move in the positive t direction.

We say that ''w'' is a ''billiard scale'' if any appropriate line in generic position, (''a''1, ..., ''a''''r'')t + ''v''0, has intersections with coordinate level planes ''x''''i'' = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.

[[Category:Fokker block]]

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 ==== Definitions: Billiard scale ====
 Let
-* w be a scale word with signature a<sub>1</sub>X<sub>1</sub>, ..., a<sub>r</sub>X<sub>r</sub>;
+* ''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);
-* n = a<sub>1</sub> + ... + a<sub>r</sub> be the length of w;
+* ''n'' = ''a''<sub>1</sub> + ... + ''a''<sub>''r''</sub> be the length of ''w'';
-* 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>.
+* ''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>i</sub>/α<sub>j</sub> for i ≠ j are irrational.
+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.
-We say that ''w'' is a ''billiard scale'' if any appropriate line in generic position, (a<sub>1</sub>, ..., a<sub>r</sub>)t + v<sub>0</sub>, has intersections with coordinate level planes x<sup>i</sup> = k that spell out the scale as you move in the positive t direction.
+We say that ''w'' is a ''billiard scale'' if any appropriate line in generic position, (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)t + ''v''<sub>0</sub>, has intersections with coordinate level planes ''x''<sup>''i''</sup> = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.
 [[Category:Fokker block]]