This is a “hint” for exercise 3.4. An infinite 2-free 3-sequence is sometimes called a Thue sequence, since the number theorist Axel Thue was the first to study them. There are several ways of generating Thue sequences. I mention three:
- One could define a map as in the case of 3-free 2-sequences. Now set and and once again consider the iterates
- Thue’s original example was and
- Another approach consists on taking the transformation giving the 3-free 2-sequence, so and Now define for to be the string obtained from by counting ones between consecutive zeros. For example, so while so Check that each is a 3-sequence. Now check that the contain no string of the form where and and conclude from this that the strings are 2-free.
If you need extra time, you have until Friday, September 11, to work on this question.