(If anyone has a better title please change it!)


Given two finite words $v,w$ in the alphabet $\{a,b\}$, define the $v$-proportion of $w$ to be the largest number of letters in $w$ which can be covered by (not necessarily disjoint) copies of the word $v$ divided by the length of $w$. Denote this quantity by $Pr(w;v)$.


As an example, $Pr(a^7ba^7b;a^4)=7/8$, but $Pr(a^7ba^7b;a^8)=0$.


I would like to find a family of finite words $w\_i$ in the alphabet $\{a,b\}$ with the following properties:


For every $n$, $\limsup\_i Pr(w\_i,a^n)=\alpha\_n>0$ but $\limsup\_n \alpha\_n=0$.



Covering sequences of words
Try $w\_0 = b$, $w\_1 = bab$, $w\_2 = baba^2bab$, $w\_3 = baba^2baba^3baba^2bab$, and recursively $w\_{i+1} = w\_ia^{i+1}w\_i$.


Then $\displaystyle \limsup\_{i \rightarrow \infty} Pr(w\_i,a^n) = \lim\_{i \rightarrow \infty} \frac{2^{i + 2 - n} +n - i -3}{3\cdot 2^i - i - 2} = \frac{2^{2-n}}{3} \rightarrow\_n 0$.


-Danny "Likes Anything That Resembles a Zimin Word" Rorabaugh


