Rank-3 scale theorems: Difference between revisions
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==== Definitions: Billiard scale ==== | ==== Definitions: Billiard scale ==== | ||
Let n = a_1 + ... + a_r be the scale | Let n = a_1 + ... + a_r be the scale length, 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. | ||
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 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. | ||