In a nutshell, the time hierarchy theorems say that a Turing machine can solve more problems if it has more time for computation. In detail for deterministic TM and time-constructable functions $f,g$ with $f(n) \log f(n) = o(g(n))$ it is $$ DTIME(f(n)) \subsetneq DTIME(g(n))$$ and for nondeterministic TM and time-constructable functions $f,g$ with $f(n+1)=o(g(n))$ it is $$ NTIME(f(n)) \subsetneq NTIME(g(n)).$$ There are a lot of (old and current) results which use the time hierarchy theorems to prove lower bounds. Here are my questions:
What happens if we can prove a better result for the deterministic or nondeterministic case?
If we can prove that there is a gap between the deterministic time hierarchy and the nondeterministic time hierarchy, does this imply $P \neq NP$?