New bounds on antipowers in words

Abstract Fici et al. defined a word to be a k-power if it is the concatenation of k consecutive identical blocks, and an r-antipower if it is the concatenation of r pairwise distinct blocks of the same size. They defined N ( k , r ) as the smallest l such that every binary word of length l contains either a k-power or an r-antipower. In this note we obtain some new upper and lower bounds on N ( k , r ) . We also consider avoiding 3-antipowers and 4-antipowers over larger alphabets, and obtain a lower bound for N ( k , 5 ) in the binary case.