Three-distance sequences with three symbols. (Q1411742)

A sequence \(S\) on an alphabet \(\{0,1,2\}\) is a three-distance sequence if, for each \(k\geq1\), \(| w|_i\) takes on no more than three values as \(w\) runs through the blocks of length \(k\) in \(S\) (here, \(| w|_i\) is the number of occurences of symbol \(i\) in the block \(w\)). \textit{W. F. Lunnon} and \textit{P. A. B. Pleasants} [J. Aust. Math. Soc., Ser. A 53, 198--218 (1992; Zbl 0759.11005)] characterized the analogously defined two-distance sequences on two symbols as being the two-dimensional cutting sequences (these are sequences of symbols recording the combinatorics of incidence of a single line on a grid). In this paper, the author shows that every three-dimensional cutting sequence is a three-distance sequence, and that there are uncountably many periodic or aperiodic three-distance sequences on three symbols which are not three-dimensional cutting sequences.