# Avoiding Squares and Overlaps Over the Natural Numbers

- Published in 2009
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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} }