Rank-3 scale theorems: Difference between revisions
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Label the notes (1,k) and (2,k), 1 ≤ k ≤ m or m-1, for notes in the upper and lower chain respectively. | Label the notes (1,k) and (2,k), 1 ≤ k ≤ m or m-1, for notes in the upper and lower chain respectively. | ||
In case 1 (even scale size), | In case 1 (even scale size), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Consider the sizes of the n/2-step (which is an odd number of generator steps): | ||
# from g1 ... g1, get a1 = (n/2-1)*g0 + g1 = n/2 g1 + (n/2-1) g2 | # from g1 ... g1, get a1 = (n/2-1)*g0 + g1 = n/2 g1 + (n/2-1) g2 | ||
# from g2 ... g2, get a2 =(n/2-1)*g0 + g2 = (n/2-1) g1 + n/2 g2 | # from g2 ... g2, get a2 =(n/2-1)*g0 + g2 = (n/2-1) g1 + n/2 g2 | ||