The time hierarchy theorem states that
$DTIME(f(n)) \neq DTIME(f(n)\log(f(n)))$
(Let me acknowledge that this statement isn't 100 percent accurate because f must be "time constructible" and potentially other things that I'm missing)
If we were to take some function f, we can consider
$DTIME(f(n)) \subset DTIME(f(n)\sqrt{\log(f(n))})\subset DTIME(f(n)\log(f(n)))$
Where we know that at least one of these inclusions is strict. Do we know of examples where one of these inclusions isn't?
In general we might want to prove that $DTIME(f(n)) \neq DTIME(f(n)g(f(n)))$ for a function $g$ which is as small as possible. Do we have better upper bounds on $g$ (like $g(n) = \log\log n$)? do we have non-trivial lower bounds on $g$ in the form of some (time constructible) $f$ for which $DTIME(f(n) = DTIME(f(n)g(f(n)))$?