Avoiding Fractional Powers over the Natural Numbers

  • Lara Pudwell
  • Eric Rowland
Keywords: Combinatorics on words, Fractional powers, Morphic sequences, Regular sequences, Automatic sequences

Abstract

We study the lexicographically least infinite a/b-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least 7/4-power-free word is a fixed point of a 50847-uniform morphism. We identify the structure of the lexicographically least a/b-power-free word for three infinite families of rationals a/b as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving a/b-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers a/b. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least 27/23-power-free word is in fact a word on the finite alphabet {0,1,2}, and its sequence of letters is 353-automatic.

Published
2018-05-25
Article Number
P2.27