On repetition-free binary words of minimal density

On Repetition-Free Binary Words of Minimal Density

Polynomial versus exponential growth in repetition-free binary words

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 73 . More precisely, there are only polynomially many binary words of length n that avoid 7 3 -powers, but there are exponentially many binary words of length n that avo...

Cyclically repetition-free words on small alphabets

All sufficiently long binary words contain a square but there are infinite binary words having only the short squares 00, 11 and 0101. Recently it was shown by J. Currie that there exist cyclically square-free words in a ternary alphabet except for lengths 5, 7, 9, 10, 14, and 17. We consider binary words all conjugates of which contain only short squares. We show that the number c(n) of these ...

Minimal Letter Frequency in n-th Power-Free Binary Words

We show that the minimal proportion of one letter in an n-th power-free binary word is asymptotically 1=n. We also consider a generalization of n-th power-free words deened through the notion of exponent: a word is x-th power-free for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in an x-th power-free binary word as a function of ...

Growth of repetition-free words -- a review

This survey reviews recent results on repetitions in words, with emphasis on the estimations for the number of repetition-free words. © 2005 Published by Elsevier B.V.