Rank-3 scale theorems: Difference between revisions

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For theorems relating to the AG property, see [[AG]].

==== Definitions: LQ ====

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

Let n = a_1 + ... + a_r be the scale size, w a scale word with signature a_1 X_1, ..., a_r X_r, let 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 contains a point (0, α_1, α_2, ... α_{r-1}) where α_i and α_i/α_j for i != j are irrational.

* Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.

Say that an r-step scale ''S'' 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 that spell out the scale as you move in the positive t direction.

* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode ''M'' of ''S'' satisfies that γ(''M'') = the graph of floor(b/a*x).

* Say that an r-step scale ''S'' is ''LQ'' if any appropriate line in generic position, (a_1, ..., a_r)t + v_0, has intersections with coordinate level planes x_i = k that spell out the scale as you move in the positive t direction.

<!--===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====

Assume wlog there are more L's than s's.

@@ Line 29: / Line 29: @@
 For theorems relating to the AG property, see [[AG]].
-==== Definitions: LQ ====
+==== Definitions: Billiard scale ====
 Let n = a_1 + ... + a_r be the scale size, w a scale word with signature a_1 X_1, ..., a_r X_r, let 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 contains a point (0, α_1, α_2, ... α_{r-1}) where α_i and α_i/α_j for i != j are irrational.
-* Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.
+Say that an r-step scale ''S'' 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 that spell out the scale as you move in the positive t direction.
-* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode ''M'' of ''S'' satisfies that γ(''M'') = the graph of floor(b/a*x).
-* Say that an r-step scale ''S'' is ''LQ'' if any appropriate line in generic position, (a_1, ..., a_r)t + v_0, has intersections with coordinate level planes x_i = k that spell out the scale as you move in the positive t direction.
 <!--===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====
 Assume wlog there are more L's than s's.