Difference Sequences and Rich Words

A Lambda word is a right infinite defined over an infinite alphabet, which has connections with Beatty sequences and Sturmian words. The Lambda word is defined by encoding differences between ordered elements of the form $i+jx$ where $i$ and $j$ are non-negative and $x$ is irrational, $1 < x < 2$. Although the Lambda word is over an infinite alphabet, palindromes in the word are over alphabets of no more than three letters. In recent years, the notion of palindromic richness has been explored. A rich word is defined as a word that exhibits the maximal number of palindromes for its length. It has been shown that a word of length $n$ can contain at most $n+1$ distinct palindromes including the empty word. An infinite word is rich If all of its factors are rich. We show that the Lambda word is rich. Furthermore, a Lambda word may be projected onto an alphabet of three letters that preserves palindromes and non-palindromes, however, the resultant word over three letters is not rich. One example of a Lambda word derives from an array known in ancient Greece that bears on the determination of musical scales in Pythagorean tuning.

Norman Carey