Words avoiding repetitions in arithmetic progressions

Details Words avoiding repetitions in arithmetic progressions Words avoiding repetitions in arithmetic progressions

2006-08-24

arxiv.org/abs/math/0608607v2

Mathematics

Combinatorics

18 pages; Manuel Silva added as a co-author

Scientific paper

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi's construction and the paperfolding words. We extend Carpi's result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avoiding overlaps and infinite words avoiding arbitrarily large squares in arithmetic progressions of odd difference. We use these words to construct labelings of the 2-dimensional integer lattice such that any line through the lattice encounters a squarefree (resp. overlapfree) sequence of labels.

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