# Avoiding Squares and Overlaps Over the Natural Numbers

• Published in 2009
• Added on
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We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103..., the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitly-defined morphism phi : N* -> N* that generates the lexicographically least infinite overlap-free word by iteration. Furthermore, we show that for all h,k in N with h <= k, the word phi^{k-h}(h) is the lexicographically least overlap-free word starting with the letter h and ending with the letter k, and give some of its symmetry properties.

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### BibTeX entry

@article{AvoidingSquaresandOverlapsOvertheNaturalNumbers,
title = {Avoiding Squares and Overlaps Over the Natural Numbers},
abstract = {We consider avoiding squares and overlaps over the natural numbers, using a
greedy algorithm that chooses the least possible integer at each step; the word
generated is lexicographically least among all such infinite words. In the case
of avoiding squares, the word is 01020103..., the familiar ruler function, and
is generated by iterating a uniform morphism. The case of overlaps is more
challenging. We give an explicitly-defined morphism phi : N* -> N* that
generates the lexicographically least infinite overlap-free word by iteration.
Furthermore, we show that for all h,k in N with h <= k, the word phi^{\{}k-h{\}}(h)
is the lexicographically least overlap-free word starting with the letter h and
ending with the letter k, and give some of its symmetry properties.},
url = {http://arxiv.org/abs/0901.1397v1 http://arxiv.org/pdf/0901.1397v1},
author = {Mathieu Guay-Paquet and Jeffrey Shallit},
comment = {},
urldate = {2016-10-03},
archivePrefix = {arXiv},
eprint = {0901.1397},
primaryClass = {math.CO},
year = 2009,
collections = {Integerology}
}