Do disjunctive sequences eventually get palindromic at some point?

Question

I have a friend who is very interested in math and has been thinking about a problem involving disjunctive sequences. For his birthday, I would like to give him an answer to his question, either by finding an existing solution or by providing the current state of research on this topic.

I considered reaching out to my nearest math institute to speak with a math professor, but I'm unsure which field this problem belongs to or who to ask for help.

I'm not a math professional, but I'll do my best to explain the problem clearly. The question is: When reading a disjunctive sequence digit by digit, is it guaranteed that at some point it will always form a palindromic sequence (from the first digit to the n-th digit)?

I would appreciate any insights, proofs, counterexamples, or references to relevant literature on this topic. If this is a known result (either proven or disproven), I would be grateful for pointers to the appropriate sources.

Thank you in advance for your help!

Answer 1

Any disjunctive sequence of $0$'s and $1$'s will always have a palindromic initial segment: say it starts with $1$, the next digit must be $0$ to avoid an immediate palindrome, and then the next $1$ will make a palindrome $10\ldots 01$.

But in an alphabet of at least $3$ letters there are disjunctive sequences with no palindromic initial segment. For example, start with a sequence listing all $m$-letter words for $m = 1, 2, \ldots$, and interpolate sufficiently many $012$'s (if $0$, $1$, $2$ are three of the letters) that more than half of the $2$-letter substrings in each initial segment are $01$, $12$ or $20$.