Hypercubic billiard word: Difference between revisions

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* Balanced ternary billiard scales with odd length are pairwise well-formed and have a well-formed [[generator sequence]] of two terms; see [[Ternary scale theorems]].

* There exist ternary billiard scales of length up to 29 that do not have a well-formed [[generator sequence]].

* A ''d''-ary billiard scale has [[balance|block balance]] ~~at most (~~''d'' − 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying ~~this~~ bound need not be billiard scales.

* A ''d''-ary billiard scale has at most ''M''-[[balance|block balanced]]. and ''N''-[[balance|chain balanced]], for ''M'', ''N'' < ''d'' − 1.<ref name="vuillon"/><ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> However, scales on at least 3 letters satisfying the block balance bound need not be billiard scales.

* An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well.

* If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4.

* A ''d''-ary billiard sclae satisfy the following two generalizations of balance on the article [[balanced word]]:

== Determining whether a scale word is a billiard scale ==

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 * Balanced ternary billiard scales with odd length are pairwise well-formed and have a well-formed [[generator sequence]] of two terms; see [[Ternary scale theorems]].
 * There exist ternary billiard scales of length up to 29 that do not have a well-formed [[generator sequence]].
-* A ''d''-ary billiard scale has [[balance|block balance]] at most (''d'' &minus; 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying this bound need not be billiard scales.
+* A ''d''-ary billiard scale has at most ''M''-[[balance|block balanced]]. and ''N''-[[balance|chain balanced]], for ''M'', ''N'' < ''d'' &minus; 1.<ref name="vuillon"/><ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> However, scales on at least 3 letters satisfying the block balance bound need not be billiard scales.
 * An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''&minus;1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well.
 * If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4.
+* A ''d''-ary billiard sclae satisfy the following two generalizations of balance on the article [[balanced word]]:
 == Determining whether a scale word is a billiard scale ==