Rank-3 scale theorems: Difference between revisions
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==== Definitions: LQ ==== | ==== Definitions: LQ ==== | ||
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 | 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 contains a point (0, α_1, α_2, ... α_r) 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. | * 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. | ||