On repetition-free binary words of minimal density
Abstract
We study the minimal proportion (density) of one letter in $n$-th power-free binary words. First, we introduce and analyse a general notion of minimal letter density for any infinite set of words which don't contain a specified set of ``prohibited'' subwords. We then prove that for $n$-th power-free binary words the density function is $\frac{1}{n}+\frac{1}{n^3}+\frac{1}{n^4}+ {\cal O}(\frac{1}{n^5})$. We also consider a generalization of $n$-th power-free words for fractional powers (exponents): 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 $x$-th power-free binary words as a function of $x$ and prove, in particular, that this function is discontinuous at $\frac{7}{3}$ as well as at all integer points $n\geq 3$. Finally, we give an estimate of the size of the jumps.