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.
@article{10_37236_6678,
author = {Lara Pudwell and Eric Rowland},
title = {Avoiding fractional powers over the natural numbers},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/6678},
zbl = {1402.68150},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6678/}
}
TY - JOUR
AU - Lara Pudwell
AU - Eric Rowland
TI - Avoiding fractional powers over the natural numbers
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6678/
DO - 10.37236/6678
ID - 10_37236_6678
ER -
%0 Journal Article
%A Lara Pudwell
%A Eric Rowland
%T Avoiding fractional powers over the natural numbers
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6678/
%R 10.37236/6678
%F 10_37236_6678
Lara Pudwell; Eric Rowland. Avoiding fractional powers over the natural numbers. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/6678
