Generator-offset property: Difference between revisions

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[Case 3: μ = anything else from 3 to floor(n/2).

Λ2 has a chunk of βs (after the first β’) of size x = either floor(n/μ) (>= floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ3 has a chunk of γs of that same size. Λ3 also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ)-1) + 1 (Λ3 might have chunks of size floor(n/μ)-1 and floor(n/μ) instead) = 2*floor(n/μ) - 1, and at most 2*(floor(n/μ)+1) + 1 = 2*floor(n/μ) + 3 (if Λ3 has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y - x, which must be 0 or 1.

Λ2 has a chunk of βs (after the first β’) of size x = either floor(n/μ) (>= floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ3 has a chunk of γs of that same size. Λ3 also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ)-1) + 1 (Λ3 might have chunks of size floor(n/μ)-1 and floor(n/μ) instead) = 2*floor(n/μ) - 1, and at most 2*(floor(n/μ)+1) + 1 = 2*floor(n/μ) + 3 (if Λ3 has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y - x, which must be 0 or 1, since Λ3 is a mos. We thus have the following cases:

~~The size difference within~~ Λ3 ~~(since it’s~~ a mos~~) must be 0 or 1, so we~~ have the following cases:

BEGIN CASEBLOCK 3.x: [ Here chunk of Λ2 means chunk of β, and chunk of Λ3 means chunk of γ.

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 [Case 3: μ = anything else from 3 to floor(n/2).
-Λ2 has a chunk of βs (after the first β’) of size x = either floor(n/μ) (>= floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ3 has a chunk of γs of that same size. Λ3 also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ)-1) + 1 (Λ3 might have chunks of size floor(n/μ)-1 and floor(n/μ) instead) = 2*floor(n/μ) - 1, and at most 2*(floor(n/μ)+1) + 1 = 2*floor(n/μ) + 3 (if Λ3 has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y - x, which must be 0 or 1.
+Λ2 has a chunk of βs (after the first β’) of size x = either floor(n/μ) (>= floor(n/floor(n/2)) = 2) or ceil(n/μ) (= floor(n/μ) + 1). Hence Λ3 has a chunk of γs of that same size. Λ3 also has a chunk that goes over n and loops back around, which must be of size y = at least 2*(floor(n/μ)-1) + 1 (Λ3 might have chunks of size floor(n/μ)-1 and floor(n/μ) instead) = 2*floor(n/μ) - 1, and at most 2*(floor(n/μ)+1) + 1 = 2*floor(n/μ) + 3 (if Λ3 has chunks of size floor(n/μ) and ceil(n/μ)). The difference between the chunk sizes is y - x, which must be 0 or 1, since Λ3 is a mos. We thus have the following cases:
-The size difference within Λ3 (since it’s a mos) must be 0 or 1, so we have the following cases:
 BEGIN CASEBLOCK 3.x: [ Here chunk of Λ2 means chunk of β, and chunk of Λ3 means chunk of γ.