The space of all sequences is a complete metric space, with the following metric: if two sequences agree on the first n positions, then they are at distance <= 2^(-n). It is easy to see that the set of all sequences avoiding cubes is closed, so there is a unique lexicographically least element. On 2/4/17 5:12 PM, David Wilson wrote:
This follows on a recent discussion of the lexically first cubefree sequence.
I have computed what I believe to be a(0..10000) of the lexically first infinite cubefree word over alphabet {0,1} (treating 0 < 1). I started to submit this sequence, however, when I got to the comments section, I began to write
Since an infinite cubefree word over {0,1} exists, a lexically first such word must exist, so this sequence is well-defined.
Then I caught myself, because I realized the reasoning was unsound. For example, there is an infinite word over {0, 1} that includes the digit 1, but no lexically first such word. So, do we actually know that there is a lexically first cubefree word over {0, 1}?
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