Avoiding Squares and Overlaps Over the Natural Numbers

TL;DR

This paper explores methods to generate the lexicographically least infinite words over natural numbers that avoid squares and overlaps, using greedy algorithms and morphisms, with explicit constructions and symmetry analysis.

Contribution

It introduces a new morphism for generating the lexicographically least overlap-free words and analyzes their structural properties.

Findings

The square-avoiding word is the well-known ruler function.

An explicit morphism generates the lexicographically least overlap-free word.

Structural symmetry properties of the overlap-free words are characterized.

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.