Rank-3 scale theorems: Difference between revisions

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The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of the LQ property.

Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let ~~γ(w)~~ be the ~~corresponding path following the word w~~ (~~γ(w)(k~~) = ~~your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {~~(a,b,c)t ~~: 0 <= t <= n}~~, ~~i.e~~. the ~~points {L~~(~~k) = (a~~,b,c) ~~k : k ∈ {0~~, ~~...~~, ~~n}}~~.

Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let L be a line of the form L(t) = (a, b, c)t ÷ v_0, where v_0 is a constant vector. We say that L is ''in generic position'' if L intersects the yz-plane at (0, α, β) where α, β, and β/α 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 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 a k-step scale ''S'' is ''LQ'' if the ~~correspomding~~ line in generic position has intersections with coordinate level planes x = k, y = k or z = k that spell out the scale.

* Say that a k-step scale ''S'' is ''LQ'' if the corresponding line, in generic position, has intersections with coordinate level planes x = k, y = k or z = 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 26: / Line 26: @@
 The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of the LQ property.
-Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let γ(w) be the corresponding path following the word w (γ(w)(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.
+Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let L be a line of the form L(t) = (a, b, c)t ÷ v_0, where v_0 is a constant vector. We say that L is ''in generic position'' if L intersects the yz-plane at (0, α, β) where α, β, and β/α 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 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 a k-step scale ''S'' is ''LQ'' if the correspomding line in generic position has intersections with coordinate level planes x = k, y = k or z = k that spell out the scale.
+* Say that a k-step scale ''S'' is ''LQ'' if the corresponding line, in generic position, has intersections with coordinate level planes x = k, y = k or z = 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.