# Balanced Word to Balanced (Sturmian?) Sequence

Let $$E \in \{0,1\}^{n}, n\in \mathbb{N}$$, be a balanced finite word: for every two subwords $$U,V$$ of the same length, the number of $$1$$'s in $$U$$ differs from the number of $$1$$'s in $$V$$ by at most one.

• Can $$E$$ be continued to an infinite balanced sequence?
• Can $$E$$ be continued to an infinite Sturmian sequence? (A subcase of the previous question.)

Furthermore, in both cases, what is the cardinality of the set of possible extensions of $$E$$?

This came up when I tried to solve an exercise, where I have to show that Sturmian sequences are dense in the space of balanced sequences (with regard to the usual topology of symbolic dynamics) - if the the answer to the second question is "yes", than we can take any prefix of length $$n$$ of a balanced sequence, continue it to a Sturmian sequence, and get by the definition of the metric that the distance between the two sequences is $$\leq \frac{1}{2^n}$$.