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According to source page 20, theorem 1.9, it states that there is a universal Turing machine that can simulate any Turing machine with a time overhead of $O(\log n)$.

Can a universal Turing machine exists with less overhead? Can we prove that no such universal Turing machine exists?

If not, can we at least prove a lower bound on the overhead?

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    $\begingroup$ I believe it is an open question. $\endgroup$ – Yuval Filmus Apr 6 '17 at 18:27
  • $\begingroup$ @Yuval What about if we restrict a UTM to simulating another UTM. Is it then still an open question? $\endgroup$ – Alexandre H. Tremblay Apr 6 '17 at 18:30
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    $\begingroup$ This isn't really well-defined. $\endgroup$ – Yuval Filmus Apr 6 '17 at 18:30
  • $\begingroup$ related $\endgroup$ – Ariel May 2 '18 at 7:15

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