User:Arseniiv/Three-gap theorem: Difference between revisions
Created page with "This is my attempt at proving the wikipedia:three-gap theorem with added MOS-related extended facts all with a single construction, so to speak. Right now (2026-04-15) I'm looking at the interval matrix, reordered to reflect the generator chain. == Preliminaries == TODO: continued fractions are bound to get used, write on their matrix composition representation using <math>S = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}</math> and <math>A = \begin{bmatrix} 1 & 1..." |
m list was autoreplaced wrong |
||
| Line 17: | Line 17: | ||
This kind of matrix looks like an overkill because all its values repeat in diagonals parallel to the main diagonal. It starts to look less so when we mark current scale steps (select a single element in each row and column, as pitches and steps alternate). Other trivial things to note for now are: | This kind of matrix looks like an overkill because all its values repeat in diagonals parallel to the main diagonal. It starts to look less so when we mark current scale steps (select a single element in each row and column, as pitches and steps alternate). Other trivial things to note for now are: | ||
* matrices for smaller scale sizes are submatrices aligned to the top right; | |||
* likewise, after stacking another generator up, we just extend the matrix with a row and a column; | |||
* each interval size is encountered once in the last row and column. | |||
A bit less trivial is that "projecting" an element onto last row and column splits it (modulo 1): <math>\{ I_{m,N} + I_{N,n} \} = \{ (N - m + n - N) g \} = \{ (n - m) g \} = I_{m,n}</math>. The sum <math>I_{m,N} + I_{N,n}</math> is always 0 or 1 larger than <math>I_{m,n}</math>, in the former case really splitting the interval. | A bit less trivial is that "projecting" an element onto last row and column splits it (modulo 1): <math>\{ I_{m,N} + I_{N,n} \} = \{ (N - m + n - N) g \} = \{ (n - m) g \} = I_{m,n}</math>. The sum <math>I_{m,N} + I_{N,n}</math> is always 0 or 1 larger than <math>I_{m,n}</math>, in the former case really splitting the interval. | ||