## Words Avoiding Repetitions in Arithmetic Progressions

#### Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, Manuel Silva

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.