# Binary words starting with arbitrarily long squares

What is the measure of the following set of infinite binary words?

$$S=\{w\in\{0,1\}^\omega\ \text{such that},\ \text{for every}\ N\in\mathbb{N},\, w\ \text{has a prefix of the form}\ pp\ \text{with}\ |p|\ge N \}$$.

• What context does this problem arise in? Dec 14, 2020 at 12:55
• I am trying to understand what can be reasonably taken as the domain of the map $\mathcal{B}$ described in this other question: mathoverflow.net/q/377105/167834 (although the link may be not completely obvious, it is not too difficult to see that this question here is basic for the other one). Dec 14, 2020 at 13:58

## 1 Answer

For any given $$p\in\{0,1\}^*$$ (finite word), the measure of the set of $$w\in\{0,1\}^\omega$$ starting with $$pp$$ is $$2^{-2|p|}$$. So, summing over all the $$p$$ with $$|p|=L$$ (there are $$2^L$$ of them), given $$L\in\mathbb{N}$$, the measure of the set $$S_L$$ of $$w\in\{0,1\}^\omega$$ starting with $$pp$$ for some $$p$$ with $$|p|=L$$ is $$2^{-L}$$. So the measure of $$\bigcup_{L=N}^{+\infty} S_L$$ is $$\leq \sum_{L=N}^{+\infty} 2^{-L} = 2^{1-N}$$, and the measure of $$S = \bigcap_{N=0}^{+\infty} \bigcup_{L=N}^{+\infty} S_L$$ is $$0$$.