Hypercubic billiard word: Difference between revisions

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* A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.

* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.

* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): ~~removing~~ any step size results in a MOS. However, the converse is false.

* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.

** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.

* Not all billiard words over more than 2 letters are [[balance|balanced]]. (In binary scales, the term "balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are balanced are [[MV3]].<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref>

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 * A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
 * All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks; [[blackdye]] can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
-* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): removing any step size results in a MOS. However, the converse is false.
+* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false.
 ** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
 * Not all billiard words over more than 2 letters are [[balance|balanced]]. (In binary scales, the term "balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are balanced are [[MV3]].<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref>