Generator-offset property: Difference between revisions

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=== Theorem 5 (Classification of MV3 scales) ===

# A single-period MV3 is PWF or equivalent to abcba~~. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. These scales are all SV3.~~

# A single-period MV3 is either PWF, equivalent to abcba, or a "twisted" word constructed as follows:

~~# A multiperiod MV3 is either a repetition of a single-period MV3 pattern~~ or a "twisted" word constructed as follows:

## Start with a power of the mos word ''w'' that begins with a X and ending with a Z and has an even number of X's.

## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''.

## Replace every other X with Y in ''w''.

## Replace every other X with Y in ''w''. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s.

# Non-twisted MV3 scales are always SV3.

# If the scale is PWF, with one exception abacaba, there always exists some "generator" interval such that the scale can be expressed as '''two parallel chains''' of this generator which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO).

==== Proof ====

Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".

Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet". Note that billiard scales are called "balanced words" in this paper.

== Open conjectures ==

@@ Line 224: / Line 224: @@
 === Theorem 5 (Classification of MV3 scales) ===
-# A single-period MV3 is PWF or equivalent to abcba. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. These scales are all SV3.
+# A single-period MV3 is either PWF, equivalent to abcba, or a "twisted" word constructed as follows:
-# A multiperiod MV3 is either a repetition of a single-period MV3 pattern or a "twisted" word constructed as follows:
 ## Start with a power of the mos word ''w''  that begins with a X and ending with a Z and has an even number of X's.
 ## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''.
-## Replace every other X with Y in ''w''.
+## Replace every other X with Y in ''w''. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s.
+# Non-twisted MV3 scales are always SV3.
+# If the scale is PWF, with one exception abacaba, there always exists some "generator" interval such that the scale can be expressed as '''two parallel chains''' of this generator which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO).
 ==== Proof ====
-Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".
+Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet". Note that billiard scales are called "balanced words" in this paper.
 == Open conjectures ==