Generator-offset property: Difference between revisions

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'''Case 3.1:''' (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) − 1).

Since ''y'' − ''x'' = floor(''n''/μ) − 1, we have ''x'' = floor(''n''/μ) = 2 and ''y'' = 3. The chunk in Λ3 whose size is 3 is made from two chunks in Λ2 of size 1. (So Λ2 has chunks of size 1 and 2, and Λ3 has chunks of size 2 and 3.)

Since ''y'' − ''x'' = floor(''n''/μ) − 1, we have ''x'' = floor(''n''/μ) = 2 and ''y'' = 3. The chunk in Λ3 whose size is 3 is made from two consecutive chunks in Λ2 of size 1. (So Λ2 has chunks of size 1 and 2, and Λ3 has chunks of size 2 and 3.) Since chunk sizes of a mos themselves form a mos, Λ2 has more chunks of size 1 than it has chunks of size 2.

~~Λ2 has two consecutive chunks of size 1.~~ Since chunk sizes of a mos themselves form a mos, Λ2 has more chunks of size 1 than it has chunks of size 2.

Λ2 has only two chunks of size 1, Λ2[''n'' − 1] and Λ2[1], since otherwise Λ3 would have a chunk of size 1 within Λ3[1 : ''n'']. Thus Λ2 has exactly one chunk of size 2. Thus Λ2 = ββ′βββ′ββ′ and Λ3 = γγ′γγγ′γγ. Thus we have:

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 '''Case 3.1:''' (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) &minus; 1).
-Since ''y'' &minus; ''x'' = floor(''n''/μ) &minus; 1, we have ''x'' = floor(''n''/μ) = 2 and ''y'' = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.)
+Since ''y'' &minus; ''x'' = floor(''n''/μ) &minus; 1, we have ''x'' = floor(''n''/μ) = 2 and ''y'' = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two consecutive chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.) Since chunk sizes of a mos themselves form a mos, Λ<sub>2</sub> has more chunks of size 1 than it has chunks of size 2.
-Λ<sub>2</sub> has two consecutive chunks of size 1. Since chunk sizes of a mos themselves form a mos, Λ<sub>2</sub> has more chunks of size 1 than it has chunks of size 2.
 Λ<sub>2</sub> has only two chunks of size 1, Λ<sub>2</sub>[''n'' &minus; 1] and Λ<sub>2</sub>[1], since otherwise Λ<sub>3</sub> would have a chunk of size 1 within Λ<sub>3</sub>[1 : ''n'']. Thus Λ<sub>2</sub> has exactly one chunk of size 2. Thus Λ<sub>2</sub> = ββ′βββ′ββ′ and Λ<sub>3</sub> = γγ′γγγ′γγ. Thus we have: