User:Arseniiv/Three-gap theorem: Difference between revisions
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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. | ||
: '''CLAIM 1a''': Two smallest intervals among the bottom-right edge <math>(m, N)</math> and <math>(N, n)</math> are the splitting of a | : '''CLAIM 1a''': Two smallest intervals among the bottom-right edge <math>(m, N)</math> and <math>(N, n)</math> are the splitting of a ''step'' of a scale one generator less. | ||
: '''CLAIM 1b''': Only one of ''m'', ''n'' can be zero, that is, at least one new step has a size that appears already at this point. | : '''CLAIM 1b''': Only one of ''m'', ''n'' can be zero, that is, at least one new step has a size that appears already at this point. | ||
:: ''(To be proven later.)'' | :: ''(To be proven later.)'' | ||