Hi I was wondering whether there is a tighter bound on the time complexity in the following theorem:
$$ \mathrm{DSPACE}(f(n)) \subseteq \mathrm{DTIME}\left(2^{O(\log n+f(n))}\right). $$
In particular, when $ f(n) \ge \log n $,
$$ \mathrm{DSPACE}(f(n)) \subseteq \mathrm{DTIME}\left(2^{O(f(n))}\right). $$
Is this exponential time overhead asymptotically tight, or is a strictly better deterministic time bound known for simulating $f(n)$-space Turing machines?
No better bound is known, and no better bound is expected to hold.
In particular, the exponential time hypothesis implies that no better bound holds in general, and specifically for $f(n)=n$.
On the other hand, an unconditional proof of this is far out of reach of the current state of knowledge. In particular, if P = PSPACE, then there is a constant $c$ such that $\mathsf{DSPACE}(f(n))\subseteq\mathsf{DTIME}(f(n)^c)$ for every time-constructible $f(n)\ge n$ by a padding argument.
