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.

: '''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 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.

:: ''(To be proven later.)''

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:: Because of the Claim 2, one of the old steps projects into a step of another old size and into one of those intervals, which depends on if its genspan was negative or positive (equivalently, if it sat on a lower or upper diagonal).

: '''CLAIM 3b''': That means if new steps aren't the same size, that one is a new size.

:: ''~~(To be proven later~~.~~)''~~

:: For now, a meh proof from contradiction: if the new steps are unequal and both happened before, then there should've been steps of both sizes in a scale one generator less, but there's also a third step size that was just split, so there were three step sizes in that scale, whereas we postulated just two.

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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.
-: '''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 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.
 :: ''(To be proven later.)''
@@ Line 35: / Line 35: @@
 :: Because of the Claim 2, one of the old steps projects into a step of another old size and into one of those intervals, which depends on if its genspan was negative or positive (equivalently, if it sat on a lower or upper diagonal).
 : '''CLAIM 3b''': That means if new steps aren't the same size, that one is a new size.
-:: ''(To be proven later.)''
+:: For now, a meh proof from contradiction: if the new steps are unequal and both happened before, then there should've been steps of both sizes in a scale one generator less, but there's also a third step size that was just split, so there were three step sizes in that scale, whereas we postulated just two.