What is the best known simulation result on a single tape TM? Does the result above also still hold?
We can simulate a multiple-tape TM on a single-tape TM with quadratic increase in time. The simulation time is $O(n^2)$. The quadratic increase is required since there are languages (e.g. palindromes) that require time $\Omega(n^2)$ on a single-tape DTM but can be solved in time $O(n)$ on a two-tape DTM.
In short, the result above does not work when the simulator is a single-tape TM.
For simulation of single-tape TMs on a single-tape TM (with arbitrary finite alphabet), the result holds, i.e. the simulation can be done with $\lg$ factor increase in time. See (2) and (3). The reference seems to be (6).
Is there any improvement on [HS66]? Can we simulate TMs on a two-tape TM for $f(n)$ steps in a faster way? Can we take $g(n)$ to be in $\omega(f(n))$ in place of $\omega(f(n)\lg f(n))$?
It seems that there hasn't been any improvement since that would imply a better time hierarchy theorem than currently known.
Correction: The usual hierarchy theorems are based on time complexity classes defined using single tape TMs. For $n$-tape TMs a tight result similar to the space hierarchy theorem is proven by <PERSON> in 1982 (5). The $\lg$ factor is not needed. Also see (4).
References:
1. <PERSON>, "Machine Models and Simulation", in Handbook of Theoretical Computer Science, 1990
(in particular, pp. 18-21)
2. <PERSON>, "Introduction to the Theory of Computation", 2006
(time complexity classes are defined using TMs with single-tape infinite in both directions and arbitrary finite alphabet, see pages 140 and 341)
3. <PERSON> , "Classical Recursion Theory", vol. I & II, 1989 & 1999
(definitions are similar to <PERSON>, see Def. I.4.1 in vol. 1020
I page 48, Def. VII.4.1 in vol. II page 67, and Thm. VII.4.15 in vol. II page 83)
4. <PERSON>, "Classical Recursion Theory", vol. II, 1999
(page 84)
5. <PERSON>, "The Tight Deterministic Time Hierarchy", 1982
6. <PERSON>, "Computational Complexity of One-Tape Turing Machine Computations", 1968
7. <PERSON> and <PERSON>, "Two-Tape Simulation of Multitape Turing Machines", 1966
8. Other related questions:
1. Lower bounds and class separation,
2. Justification of $\lg f$ in DTIME hierarchy theorem,
3. Alphabet of single-tape <PERSON> machine,
4. For the time hierarchy theorem, how is the input translated efficiently?,
5. a comment by <PERSON>. 489